Multiscale physical etch modeling and methods thereof

ABSTRACT

Systems and methods for simulating a plasma etch process are disclosed. According to certain embodiments, a method for simulating a plasma etch process may include predicting a first characteristic of a particle of a plasma in a first scale based on a first plurality of parameters; predicting a second characteristic of the particle in a second scale based on a modification of the first characteristic caused by a second plurality of parameters; and simulating an etch characteristic of a feature based on the first and the second characteristics of the particle. A multi-scale physical etch model or a multi-scale data driven model may be used to simulate the plasma etch process.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority of EP application 20193506.1 which wasfiled on Aug. 29, 2020 and which is incorporated herein in its entiretyby reference.

TECHNICAL FIELD

The embodiments provided herein disclose methods of modeling andsimulating semiconductor fabrication processes, and more particularlymethods for multiscale physical etch modeling and simulation to mitigateprocess asymmetries in metrology and semiconductor processingtechniques.

BACKGROUND

While previous generations of integrated circuits could be fabricatedusing wet, chemical etching techniques, complex chip designs of todayand the future cannot be made without using plasma processes to obtainthe necessary pattern transfer fidelity. Plasma equipment and plasmaprocessing play a vital role in fabrication of three-dimensional devicessuch as 3D-FLASH, or 1-megabit dynamic random access memories (DRAMs).Features such as gate electrodes or interconnection vias have widthsthat are comparable to the thin film thickness, therefore, to transferthe pattern with high fidelity, etching must be anisotropic, i.e., muchfaster perpendicular than parallel to the surface.

Although plasma etching is desirable in producing patterns with highaspect ratios, it may cause etch-induced asymmetry of alignment featuresacross multiple layers in a chip, leading to etch overlay and alignmenterrors. As microelectronic devices continue to shrink and processrequirements become more stringent, plasma modeling and simulationbecomes increasingly more attractive as a tool for design, control, andoptimization of mask designs, die designs, and etch recipes. Theexisting modeling techniques including physical modeling and data-drivenmodeling, though better than experimental techniques, areresource-intensive, time consuming, non-scalable, and do not account forcrosstalk between multiple length scales. These limitations render theexisting modeling techniques inadequate and inefficient.

SUMMARY

One aspect of the present disclosure is directed to a method forsimulating a plasma etch process, the method comprising predicting afirst characteristic of a particle of a plasma in a first scale based ona first plurality of parameters, predicting a second characteristic ofthe particle in a second scale based on a modification of the firstcharacteristic caused by a second plurality of parameters, andsimulating an etch characteristic of a feature based on the first andthe second characteristics of the particle.

The method may further comprise predicting a sheath profile of theplasma in the first scale based on the first plurality of parameters,wherein the first scale comprises a wafer-scale. Predicting the firstcharacteristic comprises determining a gradient of the predicted sheathprofile, and wherein the first characteristic comprises an angle ofincidence, a trajectory, or an energy of the particle directed towards awafer. The first plurality of parameters may comprise geometry of aplasma reactor configured to perform the plasma etch process, a processcondition for the plasma etch process, or a location on the wafer.Predicting the second characteristic may comprise predicting amodification of the angle of incidence, the trajectory, or the energy ofthe particle in the second scale, and wherein the second scale comprisesa die-scale. Predicting the second characteristic may further compriseaccessing a layout of a die, the layout comprising a pattern densitymap, and predicting the second characteristic of the particle based onthe pattern density map, wherein the particle may comprise a chargedparticle or an uncharged particle. Predicting the second characteristicof the charged particle of the plasma may further comprise identifying,based on the pattern density map, a first region of the die having afirst pattern density and a second region of the die having a secondpattern density different from the first pattern density; and predictingan electric potential gradient between the identified first and thesecond regions; and predicting the second characteristic of the chargedparticle based on the electric potential gradient. In some embodiments,predicting the second characteristic of the uncharged particle of theplasma may comprise predicting a concentration gradient of an etchantbetween the identified first and the second regions, predicting adiffusion flux of the etchant based on the concentration gradient, andpredicting the second characteristic of the uncharged particle based onthe diffusion flux. The method may further comprise predicting themodified trajectory of the particle in the second scale based on apattern density gradient of the die and a Gaussian kernel, wherein theGaussian kernel is a multi-length scale kernel comprising a length scaleranging from 5 nm to 50 μm. The second plurality of parameters maycomprise the layout, the pattern density, or a pattern density variationof the die. Simulating the etch characteristic may comprise simulatingan etch rate, an etch profile, or an etch asymmetry of the feature basedon the pattern density map of the die. In some embodiments, patterndensity is characterized or represented pattern perimeter density.

Another aspect of the present disclosure is directed to a method forgenerating a simulated image of a feature. The method may compriseacquiring a first image of the feature, identifying the feature based ona pattern or pattern-perimeter information from the image, andpredicting an etch profile of the feature to be etched using a plasmaetch process. Predicting the etch profile of the feature may comprisepredicting a first characteristic of a particle of a plasma in a firstscale based on a first plurality of parameters, and predicting a secondcharacteristic of the particle in a second scale based on a modificationof the first characteristic caused by a second plurality of parameters.The method may further comprise generating a second image comprising thepredicted etch profile of the feature.

Another aspect of the present disclosure is directed to a plasma etchsimulation system, comprising a memory storing a set of instructions,and a processor configured to execute the set of instructions to causethe plasma etch simulation system to predict a first characteristic of aparticle of a plasma in a first scale based on a first plurality ofparameters, predict a second characteristic of the particle in a secondscale based on a modification of the first characteristic caused by asecond plurality of parameters, and simulate an etch characteristic of afeature based on the first and the second characteristics of theparticle.

The processor may be configured to execute the set of instructions tofurther cause the plasma etch simulation system to predict a sheathprofile of the plasma in the first scale based on the first plurality ofparameters, determine a gradient of the predicted sheath profile; anddetermine an angle of incidence, a trajectory, or an energy of theparticle directed towards a wafer based on the gradient of the predictedsheath profile. In some embodiments, the processor may be configured toexecute the set of instructions to further cause the plasma etchsimulation system to access a layout of a die, the layout comprising apattern density map, e.g., a pattern-perimeter density map, and predictthe second characteristic of the particle based on the pattern densitymap, wherein the particle may comprise a charged particle or anuncharged particle. In some embodiments, the processor may be configuredto execute the set of instructions to further cause the plasma etchsimulation system to identify, based on the pattern density map, a firstregion of the die having a first pattern density and a second region ofthe die having a second pattern density different from the first patterndensity, predict an electric potential gradient based on the identifiedfirst and the second regions, and predict the second characteristic ofthe charged particle based on the electric potential gradient. In someembodiments, the processor may be configured to execute the set ofinstructions to further cause the plasma etch simulation system topredict a concentration gradient of an etchant between the identifiedfirst and the second regions, predict a diffusion flux of the etchantbased on the concentration gradient, and predict the secondcharacteristic of the uncharged particle based on the diffusion flux.

Another aspect of the present disclosure is directed to a non-transitorycomputer readable medium storing a set of instructions that isexecutable by one or more processors of an apparatus to cause theapparatus to perform a method of simulating a plasma etch process. Themethod may comprise predicting a first characteristic of a particle of aplasma in a first scale based on a first plurality of parameters,predicting a second characteristic of the particle in a second scalebased on a modification of the first characteristic caused by a secondplurality of parameters, and simulating an etch characteristic of afeature based on the first and the second characteristics of theparticle.

Another aspect of the present disclosure is directed to a non-transitorycomputer readable medium storing a set of instructions that isexecutable by one or more processors of an apparatus to cause theapparatus to perform a method of simulating a plasma etch process. Themethod may comprise acquiring a first image of the feature, identifyingthe feature based on a pattern-perimeter information from the image, andpredicting an etch profile of the feature to be etched using a plasmaetch process. Predicting the etch profile of the feature may comprisepredicting a first characteristic of a particle of a plasma in a firstscale based on a first plurality of parameters, and predicting a secondcharacteristic of the particle in a second scale based on a modificationof the first characteristic caused by a second plurality of parameters.The method may further comprise generating a second image comprising thepredicted etch profile of the feature.

Another aspect of the present disclosure is directed to a method ofsimulating a plasma etch process. The method may include predicting, ina first scale, a first characteristic of a chamber of a plurality ofchambers configured to perform the plasma etch process, predicting, in asecond scale, a second characteristic of the chamber of the plurality ofchambers, wherein the first scale comprises the second scale, andsimulating an etch characteristic of a feature based on the first andthe second characteristics of the chamber.

Another aspect of the present disclosure is directed to a plasma etchsimulation system. The system may include a memory storing a set ofinstructions, and a processor configured to execute the set ofinstructions to cause the plasma etch simulation system to predict, in afirst scale, a first characteristic of a chamber of a plurality ofchambers configured to perform the plasma etch process, predict, in asecond scale, a second characteristic of the chamber of the plurality ofchambers, wherein the first scale comprises the second scale, andsimulate an etch characteristic of a feature based on the first and thesecond characteristics of the chamber.

Other advantages of the embodiments of the present disclosure willbecome apparent from the following description taken in conjunction withthe accompanying drawings wherein are set forth, by way of illustrationand example, certain embodiments of the present invention.

BRIEF DESCRIPTION OF FIGURES

FIG. 1 is a schematic diagram illustrating an exemplary plasma processsimulation system, consistent with embodiments of the presentdisclosure.

FIG. 2 is a schematic diagram illustrating an exemplary apparatusconfigured for performing a plasma process, consistent with embodimentsof the present disclosure.

FIG. 3 illustrates a flowchart for a multi-scale plasma etch model,consistent with embodiments of the present disclosure.

FIGS. 4A and 4B illustrate a top view and a cross-section view,respectively, of a wafer configured to be exposed to a plasma,consistent with the embodiments of the present disclosure.

FIG. 5 illustrates a schematic of a die located at a radial distance rfrom the center of the wafer, consistent with embodiments of the presentdisclosure.

FIG. 6 illustrates a deviation of ion trajectory at a junction betweenthe dense and isolated regions of a die, consistent with embodiments ofthe present disclosure.

FIG. 7A illustrates a schematic of a dense region and an isolated regionof a die, consistent with embodiments of the present disclosure.

FIGS. 7B and 7C illustrate a plot of simulated differential surfacecharging effects on dense and isolated regions of a die, consistent withembodiments of the present disclosure.

FIG. 8A illustrates a schematic of a region of a die comprising a highpattern-perimeter density region, a low pattern-perimeter densityregion, and a junction region, consistent with embodiments of thepresent disclosure.

FIG. 8B illustrates a plot of a simulated ion angular distribution on ahigh pattern-perimeter density region of a die, consistent withembodiments of the present disclosure.

FIG. 9 illustrates a plasma simulation system comprising a data-drivenmodel for simulating a plasma etch process, consistent with embodimentsof the present disclosure.

FIG. 10 illustrates a process flowchart of an exemplary simulationmethod for simulating a plasma etch process, consistent with embodimentsof the present disclosure.

FIG. 11 illustrates a process flowchart of an exemplary simulationmethod for simulating a plasma etch process, consistent with embodimentsof the present disclosure.

FIG. 12 illustrates an exemplary multi-scale physical plasma etch model,consistent with embodiments of the present disclosure.

DETAILED DESCRIPTION

Reference will now be made in detail to exemplary embodiments, examplesof which are illustrated in the accompanying drawings. The followingdescription refers to the accompanying drawings in which the samenumbers in different drawings represent the same or similar elementsunless otherwise represented. The implementations set forth in thefollowing description of exemplary embodiments do not represent allimplementations. Instead, they are merely examples of apparatuses andmethods consistent with aspects related to the disclosed embodiments asrecited in the appended claims. For example, although some embodimentsare described in the context of utilizing electron beams, the disclosureis not so limited. Other types of charged particle beams may besimilarly applied. Furthermore, other imaging systems may be used, suchas optical imaging, photo detection, x-ray detection, etc.

Electronic devices are constructed of circuits formed on a piece ofsilicon called a substrate. Many circuits may be formed together on thesame piece of silicon and are called integrated circuits or ICs. Thesize of these circuits has decreased dramatically so that many more ofthem can fit on the substrate. For example, an IC chip in a smart phonecan be as small as a thumbnail and yet may include over 2 billiontransistors, the size of each transistor being less than 1/1000th thesize of a human hair.

Making these extremely small ICs is a complex, time-consuming, andexpensive process, often involving hundreds of individual steps. Errorsin even one step have the potential to result in defects in the finishedIC, thereby rendering it useless. Thus, one goal of the manufacturingprocess is to avoid such defects to maximize the number of functionalICs made in the process, that is, to improve the overall yield of theprocess.

Low pressure, cold, weakly ionized glow discharge plasmas are usedextensively in the processing of semiconductor materials. Plasmas may beused for etching and deposition of thin films of semiconductors anddielectric materials. The goals of a plasma etch process are to achievehigh etch rate, uniformity, selectivity, anisotropy, and no radiationdamage. High etch rate is desirable to increase the process throughput,however, etch rate must be balanced against uniformity, selectivity, andanisotropy. There are a number of externally controlled variables(process inputs) that may influence the plasma characteristics, and inturn the process output. For a given etch chamber configuration, theetch conditions such as plasma power, pressure, frequency, etc. may beadjusted to affect the etch rates, etch uniformity, etch selectivity, orthe like. Although plasma process development has been largely based onexperimental procedures, as devices continue to become increasinglycomplex, computer-aided design of plasma processes based on modeling andsimulation have become more attractive.

Existing physical etch models may be able to predict plasmacharacteristics and etch profiles, but are limited to device featurescales and lack scalability. Extrapolating device feature scalesimulations to wafer scale simulations may be extremely inefficient andinaccurate due to computational constraints. Existing data-driven modelsincluding convolutional encoder-decoder networks, neural networks, deeplearning algorithms, etc. may require intensive training for eachprocess, multiple locations on the wafer, and reticle designs. Further,neither the existing physical etch models nor data-driven models accountfor crosstalk between length scales ranging from feature scale to waferscale, spanning 6 to 8 orders of magnitude of device dimensions.

Some embodiments of the present disclosure are directed to methods ofsimulating a plasma etch process using a multi-scale plasma etch model.The method includes predicting, using a wafer-scale model, thecharacteristics of an etchant ion species in a wafer-scale based onplasma etch chamber geometry or plasma etch process conditions at alocation on the wafer. The method may further include predicting, usinga die-scale model, the modification of characteristics of the etchantion species in die-scale based on a multiscale gradient of a Gaussiankernel convolved with pattern density of the die. In some embodiments,patterns density is represented or characterized by pattern-perimeterdensity of the die. The information obtained from the wafer-scale modeland the die-scale model are used as input to a feature scale etch model,either physical or data-driven, to simulate the etch profile of afeature on the die. The method provides a multi-scale physical or adata-driven etch model to simulate etch profiles, useful in mitigatingetch-induced asymmetries and etch-induced overlay errors.

Relative dimensions of components in drawings may be exaggerated forclarity. Within the following description of drawings, the same or likereference numbers refer to the same or like components or entities, andonly the differences with respect to the individual embodiments aredescribed. As used herein, unless specifically stated otherwise, theterm “or” encompasses all possible combinations, except whereinfeasible. For example, if it is stated that a component may include Aor B, then, unless specifically stated otherwise or infeasible, thecomponent may include A, or B, or A and B. As a second example, if it isstated that a component may include A, B, or C, then, unlessspecifically stated otherwise or infeasible, the component may includeA, or B, or C, or A and

B, or A and C, or B and C, or A and B and C.

Reference is now made to FIG. 1 , which illustrates an exemplary plasmaprocess simulation system 100, consistent with embodiments of thepresent disclosure. As shown in FIG. 1 , plasma process simulationsystem 100 may comprise a plasma process simulation system including anapparatus 105 in direct or indirect communication with a processor 180,and a controller 150 configured to control apparatus 105. Apparatus 105may communicate with processor 180 wirelessly, remotely, or through awired connection, among other communication methods. Apparatus 105 mayinclude a plasma process chamber 110, a gas supply system 120 to supplyand regulate gases or a mixture of gases to plasma process chamber 110,a vacuum system 140, and a power supply 160. While the description anddrawings are directed to ionized gases, it is appreciated that theembodiments are not used to limit the present disclosure to specificcharged particles.

In the field of semiconductor fabrication and processing,plasma-assisted material processes are generally carried out in plasmachambers or plasma reactors, such as plasma process chamber 110. Plasmareactors, based on the method of excitation of plasma and chamberconfiguration, may be used to perform processes including, but notlimited to, etching, deposition, surface treatment, or defect detectionin wafers (e.g., semiconductor wafers or wafers made of othermaterial(s)) or samples (wafers and samples are collectively referred toas “wafers” or “wafer” hereafter).

Capacitively Coupled Plasma (CCP) reactors, and high density plasmareactors such as Inductively Coupled plasma (ICP) reactors and ElectronCyclotron Resonant (ECR) plasma reactors, have been widely used in thesemiconductor industry for Plasma Enhanced Chemical Vapor Deposition(PECVD) and Reactive Ion Etching (RIE) or plasma-assisted high-aspectratio etching. A conventional plasma reactor, such as a CCP reactor,typically consists of two parallel plate electrodes in a chamber. Thereactive nature of the discharged gas in the chamber is sustained due tothe radio frequency (RF) voltage on the two electrodes and the highvoltage on the electrodes causes ion bombardment on the surface of awafer. The typical pressure in the chamber ranges from 10-3-10 Torr. AnICP reactor typically consists of two sets of RF coils located outsidethe plasma chamber. RF power is provided to the chamber by an inductivemagnetic field. In general, by adjusting RF bias voltage to the wafer,one can independently control the ion bombardment energy.

Gas supply system 120 may be configured to supply and regulate thesupply of gases used in the plasma process to plasma process chamber110. Gas supply system 120 may include gas flow controllers, gas flowmonitors, gas mixers, gas manifold, gas lines, among other components tohelp control the flow rate, concentration, proportion of gases suppliedto plasma process chamber 110. Gas supply system 120 and plasma processchamber 110 may be controlled by controller 150, which controls andregulates the introduction of various gases and carrier gas to plasmaprocess chamber 110. A carrier gas may include an inert gas or a mixtureof inert gases that may be used to “carry” or deliver desired activegases. The carrier gas may not react with the active gases or with theby-products of the plasma process.

Plasma process chamber 110 may be connected to vacuum system 140, whichremoves gas molecules in plasma process chamber 110 to reach a firstpressure below the atmospheric pressure. Vacuum system 140 may includemore than one vacuum pump such as, but not limited to, a mechanicalpump, a diffusion pump, a turbomolecular pump, an ion pump, or acombination thereof, to obtain the desired first pressure. Afterreaching the first pressure, desired gases may be introduced into plasmaprocess chamber 110. The introduction of desired gases may raise thepressure in plasma process chamber 110 to a desired second pressure,typically in the range of 1 mTorr to 10 Torr. After reaching the secondpressure, the wafer may be exposed to a plasma generated in plasmaprocess chamber 110 based on the chemistry of gases, flow rate,configuration, power distribution, among other factors. Vacuum system140 may be controlled by controller 150 to, for example, adjust thechamber pressure by adjusting valve positions, valve timing, among otherthings.

Plasma process chamber 110 may be connected to power supply 160, whichis configured to supply and regulate power to one or more electrodes,coils, or the like. In an example, power supply 160 may be configured toapply RF voltage to an electrode, while maintaining another electrode ata reference voltage, to generate an alternating electric field betweentwo electrodes. The alternating electric field may be utilized to excitethe gas molecules, thereby generating a plasma in plasma process chamber110. It is appreciated that power supply 160 may be used to supplyelectrical power to one or more components of apparatus 105, asappropriate.

Controller 150 may be electronically connected to apparatus 105 and maybe electronically connected to other components including, but notlimited to, processor 180, as well. Controller 150 may be a computerconfigured to execute various controls of apparatus 105 using processingcircuitry to execute various signal and data processing functions. Whilecontroller 150 is shown in FIG. 1 as being outside of the structure thatincludes plasma process chamber 110, it is appreciated that controller150 may be a part of apparatus 105.

Processor 180 may be a computer configured to communicate withcontroller 150 or apparatus 105. As shown, processor 180 may communicatewith apparatus 105 through controller 150. In an example wherecontroller 150 is a part of apparatus 105, processor 180 may directlycommunicate with apparatus 105. Processor 180 may include a memory tostore a set of instructions which, upon execution, may allow plasmaprocess chamber 110 to perform a desired function. In some embodiments,processor 180 may be configured to receive instructions from a userthrough a user-interface, perform simulation and mathematical modelingof a process based on user input, predict a process outcome, andgenerate an image depicting the predicted process outcome.

Reference is now made to FIG. 2 , which illustrates an exemplaryapparatus 200 configured for performing a plasma process, consistentwith embodiments of the present disclosure. Apparatus 200 may include aplasma process chamber 210, a gas supply system 220, a stage positioningsystem 230, a vacuum system 240, a controller 250, a power supply systemcomprising power generators 260 a and 260 b, and a pressure sensor 270.It is to be appreciated that other relevant components may be added oromitted, as needed.

While the present disclosure provides examples of plasma process chamber210 configured to be used as a plasma etching system, it should be notedthat aspects of the disclosure in their broadest sense, are not limitedto a plasma etch chamber or an etching system. Rather, it is appreciatedthat the foregoing principles may be applied to other chambers as well.For example, plasma process chamber 210 may be configured for use as adeposition chamber to grow thin films of semiconductors or dielectrics,or as a surface treatment chamber to strip residual photoresist.

In some embodiments, plasma process chamber 210 may be configured foruse as a plasma etch chamber or a plasma etch reactor, and therefore,may also be referred to hereafter as a plasma etch chamber 210. In anexemplary plasma etch process, a wafer such as wafer 203 may be placedin plasma etch chamber 210 such that wafer 203 may be exposed to aplasma generated by introduction of an etchant gas or an etchant gasmixture. Plasma etch chamber 210 may include a gas supply system 220that may deliver one or more gaseous etchants to plasma etch chamber210. The gas supply system 220 may be configured to supply the variousdesired gaseous etchants to plasma etch chamber 210 through a gascontroller 228 and a feedline 229. In some embodiments, a gas supplysystem 220 may also be configured to control the flow rate of an etchantgas or a mixture of etchant gases into plasma etch chamber 210 bycontrolling the flow and pressure of a carrier gas through gas supplysystem 220. In some embodiments, the etching process performed by plasmaetch chamber 210 may be a RIE or deep reactive-ion etching (DRIE)process.

In some embodiments, gas supply system 220 may include gas sources 222,224, and 226. In an exemplary embodiment, gas sources 222 and 224 maycomprise etchant gases, and gas source 226 may comprise a carrier gas.Although only three gas sources 222, 224, and 226 are illustrated inFIG. 2 , this is done merely for clarity, and it should be appreciatedthat any suitable number of etchant gas sources may be included. Forexample, in an embodiment in which five separate etchants may beutilized, there may be five etchant gas sources, or four etchant gassources and one carrier gas source, or three etchant gas sources and twocarrier gas sources, or other configurations may be possible as well, asneeded.

Each of the gas sources 222, 224, and 226 may be a vessel, such as a gasstorage tank or a gas cylinder, or a gas dewar, placed locally orremotely from plasma etch chamber 210. In some embodiments, gas supplysystem 220 may be part of a facility that independently prepares anddelivers the desired etchants. Any suitable source for the desiredetchants may be utilized as a gas source, and all such sources are fullyintended to be included within the scope of the embodiments. A carriergas or a diluent gas may be used to help push or “carry” the variousdesired etchants to plasma etch chamber 210. The carrier gas mayinclude, but is not limited to, Nitrogen (N₂), Helium (He), Argon (Ar),Xenon (Xe), or a combination thereof, or other suitable carrier gasesmay be utilized as well.

As illustrated in FIG. 2 , gas supply system 220 may include gas flowvalves 223, 225, and 227 configured to regulate the flow rate, flowamount, or flow direction of etchant gases from gas sources 222, 224,and 226, respectively, to gas controller 228. In some embodiments, gascontroller 228 may be configured to combine the various etchants andcarrier gases to prepare gas mixtures having a predefined proportion ofgases, and once combined, the gas mixture may be directed towards plasmaetch chamber 210 through feedline 229. Although, each gas source 222,224, and 226 are shown connected to gas controller 228, a gas source maybe separately and directly connected to plasma etch chamber 210.Controller 250 may be configured to control one or more functions of gassupply system 220. For example, controller 250 may control the operationof one or more gas flow valves 223, 225, and 227, or control theoperation of gas controller 228, or other functions related to gassupply system 220, as appropriate.

An exemplary plasma etch chamber 210, as shown in FIG. 2 , may includean upper electrode 201, a gas showerhead 202, a wafer 203 mounted on astage 204, a focus ring 205 adjustable along one or more of X-, Y-, orZ-axes using a focus ring position controller 206, and a lower electrode207. The gases introduced from gas supply system 220 may be “excited” toform a plasma 208 comprising ions, free radicals, neutral species, andcharged particles. In the context of this disclosure, “exciting” a gasmixture refers to subjecting the gas to an adequate electromagneticfield to extract electrons from the gas atoms, thereby ionizing the gasand forming a plasma.

Plasma etch chamber 210 may include upper electrode 201 and lowerelectrode 207. In a capacitively coupled plasma reactor such asapparatus 200 of FIG. 2 , upper electrode 201 may be a powered electrodeand lower electrode 207 may be an earthed or a grounded electrode. Insome embodiments, upper electrode 201 may be grounded and lowerelectrode 207 may be powered. In some embodiments both upper electrode201 and lower electrode 207 may be powered. Ions in the plasma may beaccelerated toward the powered electrode and the potential differencebetween the plasma and the powered electrode is generally referred to asbias voltage. The configuration may be modified based on theapplication, or a desired mode of etching. For example, in the RIE mode,wafer 203 may be placed on a powered electrode and may experience thebias voltage. The reactive ions as well as other reactive species fromthe plasma may cause etching of a feature on wafer 203. In the plasmaetch mode, wafer 203 may be placed on a grounded electrode and thereactive neutral species of the plasma may cause the etching.

Upper electrode 201 and lower electrode 207 may comprise electricallyconductive electrodes that can be electrically biased with respect toeach other to generate an electric field strong enough to ionize gasesbetween the electrodes into a plasma. In some embodiments, upperelectrode 201 or lower electrode 207 may be configured to receive anelectric charge. Electrical power, generally in the form of ahigh-frequency (radio frequency of 13.56 MHz) RF power may be applied toupper electrode 201, lower electrode 207, or both, using powergenerators 260 a and 260 b. The powered upper electrode 201 mayfacilitate a uniform distribution of plasma 208 in the plasma excitationregion between wafer 203 and gas showerhead 202. In some embodiments,one or both power generators 260 a and 260 b may be electrically coupledto upper electrode 201 and lower electrode 207, respectively, to deliveran adjustable amount of power depending on the process performed. Forexample, in an etching process, the power delivered to the electrodesmay be adjusted to adjust the etch selectivity or etch uniformity of alayer or a feature on wafer 203.

In some embodiments, gas showerhead 202 may be configured to receive thevarious etchants from gas supply system 220 and disperse the variousetchants into plasma etch chamber 210. Gas showerhead 202 may bedesigned to evenly disperse the etchants in order to maximize uniformityof process conditions including, but not limited to, plasma coverage,plasma density, plasma intensity, plasma shape, or the like. Gasshowerhead 202 may comprise openings arranged unevenly or evenly in arectangular, a triangular, a circular, a non-circular, or a spiralpattern, or a combination thereof. It is appreciated that any suitablemethod of dispersing the desired etchants, such as entry ports, spraynozzles, or the like may be utilized to introduce the desired etchantsinto plasma etch chamber 210.

Plasma etch chamber 210 may include stage 204 configured to secure wafer203 during the etching process. In some embodiments, wafer 203 may bemounted onto a mounting surface (not shown) of stage 204. Wafer 203 maybe secured on stage 204 using electrostatic forces, mechanical clamps,vacuum pressure, or a combination thereof, and may also include heatingand cooling mechanisms configured to control the temperature of wafer203 during the processes.

In some embodiments, plasma etch chamber 210 may include a focus ring205 mounted on a focus ring holder (not shown) or stage 204. Focus ring205 may surround wafer 203 and may have a generally annular shape. Focusring 205 may have a rectangular cross-section, or may have an irregularcross-section or a cross-section of a different shape. In someembodiments, focus ring 205 may be made of a conductive material, asemiconductor material, a dielectric material, or another suitablematerial. In some embodiments, the focus ring 205 may be made of dopedor undoped silicon. The focus ring holder may be connected to a focusring position controller 206 configured to move focus ring 205vertically along Z-axis. In some embodiments, focus ring positioncontroller 206 may be utilized to control the vertical position of focusring 205 during the etching process or in some embodiments a DC voltagecan be applied to focus ring 205. Both the vertical position of focusring 205 or the DC voltage applied to focus ring 205 may affectcharacteristics of the etching process such as including, but notlimited to, etch tilt, etch uniformity, etch rate, or the like. It isappreciated that focus ring position controller 206 or focus ring DCvoltage may be configured before, during, or after the etching process.

In some embodiments, apparatus 200 may include stage positioning system230 configured to adjust a position of stage 204, thereby adjusting aposition of wafer 203 secured on stage 204, along one or more of X-, Y-,or Z-axes. Stage positioning system 230 may include, but not limited to,piezo actuators, position sensors, micro-positioners, or the like, toprecisely adjust the position of wafer 203 with respect to plasma 208,which may impact the etch characteristics of a feature on wafer 203. Forexample, adjusting the height of stage 204 such that the verticaldistance between wafer 203 and upper electrode 201 is reduced, mayaffect the etch rate, etch profile, or etch anisotropy, or other etchcharacteristics. Stage positioning system 230 may communicate withcontroller 250 to allow controller 250 to adjust the position of stage204 in X-, Y-, or Z-axes.

Apparatus 200 may further include vacuum system 240 configured to“evacuate” plasma etch chamber 210 to a predefined pressure. In someembodiments, vacuum system 240 may be configured to pump out air,moisture, residual gases, or the like, from plasma etch chamber 210prior to introducing etchant gases from gas supply system 220. In someembodiments, vacuum system 240 may be further configured to “refill”plasma etch chamber 210 with ambient gas or carrier gas from gas supplysystem 220 to bring plasma etch chamber 210 to atmospheric pressure.Vacuum system 240 may comprise one or more vacuum pumps, pressuregauges, valves, among other components, and may be controlled usingcontroller 250.

Apparatus 200 may further include pressure sensor 270 configured tomeasure gas pressure in plasma etch chamber 210 during the etchingprocess. Etchant gases may be introduced in plasma etch chamber 210 andstabilized before igniting a plasma by subjecting the introduced gasesto high potential difference between upper electrode 201 and lowerelectrode 207. Once stabilized, gas pressure may be maintained oradjusted during the etching process to adjust the etch characteristicsincluding, but not limited to, etch rate, etch selectivity, etchanisotropy, etch asymmetry, or the like. Pressure sensor 270 maycomprise a pressure gauge such as a bourdon tube pressure gauge, acapacitance manometer, a Pirani gauge, or the like. Pressure sensor 270may be controlled using controller 250 and may communicate with gassupply system 220 directly or indirectly through controller 250.

Apparatus 200 may further include controller 250 configured to controlgas supply system 220, stage positioning system 230, vacuum system 240,power generators 260 a and 260 b, pressure sensor 270, or othercomponents, as appropriate. Controller 250 may be analogous to and mayperform substantially similar functions as controller 150 of FIG. 1 .

As microelectronic devices continue to shrink and process requirementsbecome more stringent, modeling and simulation of plasma processes mayoffer more insight and accurate predictability of process outcomes,tighter process control, optimized tool design, among other advantages.In plasma etching processes used for semiconductor device fabrication,high etch rates, uniformity, selectivity, controlled shape of themicroscopic features being etched (anisotropy), or minimal radiationdamage, among other things, may be desirable. High etch rates may bedesirable to increase the process throughput. Uniformity refers toachieving the same etch characteristics (rate, profile, etc.) across thewafer. Uniformity may be desirable, for example, to minimize non-uniformelectrical charging of the wafer that can lead to electrical damage.Selectivity refers to the relative rate of etching of one material withrespect to another. Selectivity may be desirable, for example, to etchan underlying layer without etching the material of the mask (a hardmask or a soft mask). Anisotropic etching may be desirable to fabricatefeatures with high aspect ratios (>1).

One of several issues encountered in modeling the behavior of plasmaprocesses is the disparity in length and time scales of plasmaprocesses. For example, length scales range from atomistic tomicroscopic (feature widths) to macroscopic (reactor, wafer), and timescales range from picoseconds, to nanoseconds (response time ofelectrons), to microseconds (response time of ions), to severalmilliseconds for heavy species chemistry and gas residence times. Thecrosstalk between these disparate length scales may contribute toprocess asymmetry and overlay errors, among other things. Althoughexisting modeling and simulation of plasma systems may provide anunderstanding of the physiochemical processes occurring in a plasma in agiven length scale, however, the crosstalk between microscopic andmacroscopic length scales is not accounted for, rendering the modelingand simulation methods inaccurate and inadequate for their desiredpurpose. As an example, the simulation method needs to account for thetrajectory deviation of species from plasma on to the wafer due topattern density variation, and more particularly pattern-perimeterdensity variations in the die. Therefore, a multi-scale etch model maybe desirable to mitigate etch-induced asymmetry and to independentlyevaluate the effect of process parameters (wafer-scale) andpattern-perimeter density variations (die-scale) on etchcharacteristics.

Reference is now made to FIG. 3 , which illustrates a flowchart for amulti-scale plasma etch model 300, consistent with embodiments of thepresent disclosure. Multi-scale etch model 300 may comprise awafer-scale model 310, a die-scale model 320, and a feature-scale model330, to generate a simulated etch profile based on the multi-scale etchmodel approach.

Wafer-scale model 310, also referred to as the large-scale model or theplasma sheath model, may comprise a data input module 302, a dataprocessing module 305, and a post-process module 308. Wafer-scale model310 may predict plasma characteristics based on a plurality ofparameters including, but not limited to, chamber geometry, processconditions (operating parameters), and a location on the wafer. Inplasma processing, the choice of reactor design and process parameters,among other things, may impact the plasma characteristics, andresultantly the process output. Plasma characteristics may include, butare not limited to, space and time variation of electrons, ions, andneutral species densities and velocities, ion flux, ion energy, ionangular distribution, ion trajectory, ion tilt, angle of incidence,neutral flux, radical flux, among other characteristics.

Data input module 302 may be configured to provide informationassociated with the chamber geometry, process conditions (operatingparameters), and location on the wafer. Information associated withchamber geometry may include plasma etch chamber type, geometricaldimensions of the chamber, materials of construction, focus ringdimensions and focus ring material, position of the focus ring, voltageapplied to the focus ring, operating condition of the focus ring, or thelike. Information associated with the process conditions may include,but not limited to, physical etch conditions such as gas pressure,plasma power, excitation frequency, substrate voltage, gas compositionand flow rate. In some embodiments, data input module 302 may beconfigured to provide information to data processing module 305.

In some embodiments, wafer-scale model 310 may comprise data processingmodule 305 configured to receive information from data input module 302.Data processing module 305 may be further configured to process thereceived information. Processing the received information may includeperforming numerical analysis of plasma sheath dynamics. Data processingmodule 305 may predict a plasma potential profile (φ) based on chambergeometry and process conditions.

Data processing module 305 may be configured to numerically solve one ormore governing equations including the equation of motion of an ion, theequation of conservation of ion flux, and the Poisson's equation forions, at wafer-scale to obtain the plasma potential profile (φ), ionenergy, ion angular distribution, ion flux, among other things.Equations 1-3 show the governing partial differential equations (PDEs)in a plasma sheath region:

$\begin{matrix}{{{\frac{\partial n}{\partial t} + {\nabla.\left( {n\overset{\rightarrow}{u}} \right)}} = 0},} & \left( {{Eq}.1} \right)\end{matrix}$ $\begin{matrix}{{{\frac{\partial\left( {n\overset{\rightarrow}{u}} \right)}{\partial t} + {{n\left( {\overset{\rightarrow}{u}.\overset{\rightarrow}{\nabla}} \right)}\overset{\rightarrow}{u}}} = {{{- \frac{en}{m}}{\overset{\rightarrow}{\nabla}\varphi}} - {v_{m}n\overset{\rightarrow}{u}}}},{and}} & \left( {{Eq}.2} \right)\end{matrix}$ $\begin{matrix}{{{\nabla^{2}\varphi} = {\frac{e}{\varepsilon_{0}}\left( {n - {n_{0}{\exp\left( \frac{\varphi - \varphi_{0}}{T_{e}} \right)}}} \right)}},} & \left( {{Eq}.3} \right)\end{matrix}$

where, Equation 1 represents the equation for conservation of mass,Equation 2 represents the equation for conservation of momentum, andEquation 3 represents the Poisson's equation for ions.

Assuming a steady state for ion flux, equations 1 and 2 can be rewrittenas equations 4 and 5 below:

$\begin{matrix}{{{\nabla.\left( {n\overset{\rightarrow}{u}} \right)} = 0},{and}} & \left( {{Eq}.4} \right)\end{matrix}$ $\begin{matrix}{{\left( {\overset{\rightarrow}{u}.\overset{\rightarrow}{\nabla}} \right)\overset{\rightarrow}{u}} = {{{- \frac{e}{m}}{\overset{\rightarrow}{\nabla}\varphi}} - {v_{m}\overset{\rightarrow}{u}}}} & \left( {{Eq}.5} \right)\end{matrix}$

In some embodiments, data processing module 305 may be configured todetermine the ion velocity {right arrow over (u)}, plasma potential φ,and plasma sheath profile as a function of chamber geometry, differentphysical etch conditions such as plasma pressure, plasma power, biasvoltage, focus ring height, etc., and a location (r) on wafer 203.

Post-process module 308 may perform further analysis of informationobtained from data processing module 305. In some embodiments,post-process module 308 may model plasma characteristics including iontilt, etch rate, angle of incidence at a given location (r) on a wafer(e.g., wafer 203 of FIG. 2 ). In some embodiments, post-process module308 may obtain information associated with ion tilt or ion energy bytracking the trajectory of a particle such as an ion, in a givenpotential field using equation 6 below. The information associated withion tilt and ion energy at a given location r in wafer-scale may serveas an input to die-scale model 320.

$\begin{matrix}{\frac{d\overset{\_}{u}}{dt} = {\frac{q_{i}}{m_{i}}{\overset{\rightarrow}{\nabla}\varphi}}} & \left( {{Eq}.6} \right)\end{matrix}$

where q_(i) and m_(i) are the charge and mass of an ion of the plasma.

Predicting the ion tilt using wafer-scale model 310 may be inadequate insimulating the final etch profile of a feature in a die because of theprobability of crosstalk induced due to disparity in length scalesbetween the wafer-scale and die-scale. For example, diameter of a12-inch wafer is −150 mm, and average dimension of a die is −5-10 mm.Further, a trajectory of an ion as it approaches a surface of a wafer(e.g. wafer 203 of FIG. 2 ), may vary based on factors including, butnot limited to, pattern-perimeter density and gradients inpattern-perimeter density, surface energy, among other things. Forexample, the ion trajectory in the plasma sheath region may be differentfrom the ion trajectory close to the wafer surface (discussed later inreference to FIG. 6 ). In the context of this disclosure, “close” to thewafer surface may refer to a distance of 2 mm or less, 1 mm or less, 500μm or less, 200 μm or less, or 100 μm or less from the wafer surface.Therefore, a die-scale model may be desirable to model the iontrajectory, or a deviation of the ion trajectory close to the waferbased on parameters associated with a die of the wafer such aspattern-perimeter density of the die.

Die-scale model 320, also referred to as the short-scale model maycomprise a data input module 312 and sub-models 315 and 318. In someembodiments, data input module 312 may be configured to receive iontilt, or ion energy information from the wafer-scale and informationassociated with a die including, but not limited to, pattern-perimeterdensity, pattern-perimeter layout, or pattern-perimeter densityvariation of a die. In some embodiments, the information from data inputmodule 312 may be utilized by one or both sub-models 315 and 318 topredict the modification of ion trajectory, ion tilt, flux of ionspecies, neutral flux, radical flux, modified by die-scalepattern-perimeter density or pattern-perimeter density variation.

Sub-model 315 of die-scale model 320 may comprise a micro-loading modelconfigured to model the loading effects and the impact of loadingeffects on local etchant availability in die-scale based on thepattern-perimeter density, or pattern-perimeter density variation in adie. Loading effects may include micro-loading and macro-loadingeffects. Micro-loading is a die-scale phenomenon and refers to adifference in etch characteristics of a given feature located in an areaof high pattern-perimeter density (dense) compared to the same featurein an area of low pattern-perimeter density (isolated) on the same die.Generally, micro-loading is caused due to local depletion of etchantspecies in dense regions, thereby causing a diffusion of reactive ionsalong a concentration gradient within the die. Macro-loading is awafer-scale phenomenon and refers to an overall reduction in etch ratedue to overall depletion of etchant species with more exposed area toetch. For example, macro-loading effects may cause a difference in etchrates of two wafers with identical features but different etchable area.Loading effect in plasma etching of semiconductor wafers is amulti-length scale effect including micro-loading effects occurring inthe length scale of 5 nanometer (nm) to 100 nm, and macro-loadingeffects occurring in the length scale of 100 nm to a few micrometers.

In some embodiments, predicting a characteristic of an unchargedparticle such as neutrals or radicals, may include predicting aconcentration gradient of an etchant between the isolated and the denseregions. The method may further include predicting a diffusion flux ofthe etchant based on the concentration gradient. In some embodiments,the characteristic of the uncharged particle, influenced bymicro-loading effects may be predicted based on the predicted diffusionflux.

In some embodiments, predicting the concentration gradient may includepredicting a first concentration of an etchant in the dense region(higher pattern-perimeter density) and predicting a second concentrationof the etchant in the isolated region (lower pattern-perimeter density).In some embodiments, the predicted first and the second concentrationsmay be compared to determine a concentration gradient across a region ofa die. The flux or the diffusion of molecules may occur from a higherconcentration region to a lower concentration region.

Sub-model 318 of die-scale model 320 may comprise a die-scale chargingmodel, also referred to as surface charging model, configured to modelthe modification of ion tilt, modification of ion angle distribution, ormodification of ion trajectory caused by differential charging of thedie surface due to pattern-perimeter density variation. Thedirectionality difference between the positively charged ions,accelerated in the plasma sheath region, and negatively chargedelectrons builds charges on insulating materials such as the photoresistmask or underlying oxide layers. As an example, low pattern-perimeterdensity (isolated) regions may be at higher potential in comparison tohigher pattern-perimeter density (dense) regions. The surface chargingmodule predicts this imbalance in surface potential and the resultingelectric fields that, for instance, can alter the trajectory of incomingions towards the dense regions. Surface charging may vary based onfactors including, but not limited to, pattern-perimeter density,pattern-perimeter density variation, or pattern-perimeter layout.

Micro-loading or surface charging effects may be expressed as a functionof pattern-perimeter density at a given location having linearcoordinates (x_(i), y_(i)) and a multi-scale gradient. The location(x_(i), y_(i)) may be at a distance r′ from the center of the die havingcoordinates (x₀, y₀). In some embodiments, the etchant concentrationgradient may be represented as in equation 7 below:

∇C=a ₀ +Σa _(i)∇(G _(i)*ρ(r′))  (Eq. 7)

where, C is the etchant concentration, a₀ and a_(i) are constantsassociated with multi-length scales of loading effects (includingmicro-loading and macro-loading effects), ρ(r′) is the local patterndensity at location r′, G_(i) is a Gaussian kernel, and ∇(G_(i)*ρ(r′))is the multi-scale gradient of local pattern density.

In some embodiments, the electric potential gradient resulting fromdifferential surface charging effect may be represented as in equation 8below:

∇V=b ₀ +Σb _(i)∇(G _(i)*ρ(r′))  (Eq. 8)

where, V is the voltage (surface potential) at the surface of the die,b₀ and b_(i) are constants associated with multi-length scales ofsurface charging effects, ρ(r′) is the local pattern density at locationr′, G_(i) is a Gaussian kernel, and ∇(G_(i)*ρ(r′)) is the multi-scalegradient of local pattern density.

In some embodiments, the gradient can be calculated based ontime-dependent etch load. In some embodiments, the time dependent etchload is a function of pattern-perimeter map and aspect ratio dependentetch. The diffusion equation can be solved as,

${\frac{\partial C}{\partial t} - {{div}\left( {D*{{grad}(C)}} \right)}} = 0$

with the following boundary condition D*grad(C)=−q(r), where C is theconcentration of species, D is the diffusion coefficient q(r) is theetchant consumption rate obtained using multiscale gradient convolution

D×grad(C)=α₀+Σα_(i)∇(G _(i) *L(r,t)),

where G is a gaussian kernel and L(r,t) is time dependent etch load

Etch load,L(r,t)=ρ0(r)×(1+tE(a)),

and where ρ0 is the pattern-perimeter map of the mask and E(a) is aspectratio dependent etch rate and a is the aspect ratio.

Multi-scale plasma etch model 300 may further comprise feature-scalemodel 330. In some embodiments, feature-scale model 330 may beconfigured to obtain information from die-scale model 320 and simulatean etch profile, etch asymmetries, or the like based on the obtainedinformation associated with the pattern-perimeter density andpattern-perimeter density variation of the die. Feature-scale model 330may include a physical etch model 332 or a data-driven etch model 334 tosimulate after-etch profiles and etch process asymmetries based onpattern-perimeter density or pattern-perimeter density variation.

Using the multi-scale model, the etch process asymmetry at feature-scalemay be determined based on the process asymmetry predicted usingwafer-scale model 310 and the modification of characteristics of ions,neutrals or radicals in plasma based on the die-scale model 320. Themodification of characteristics may include local perturbations atfeature scale due to short length scale effects including differentialsurface charging effects, micro-loading effects. The local perturbationsmay be determined based on the pattern-perimeter density and a gradientof the pattern-perimeter density. The overall process asymmetry atfeature scale may be represented as in equation 9 below:

A(r+r′)=g((φ_(r)),ρ₀(r′),∇ρ0(r′))  (Eq. 9)

where, A is the overall process asymmetry, φ_(r) is the ion tilt at agiven location r in wafer-scale, ρ0(r′) is the local pattern-perimeterdensity at location r′, and ∇ρ0(r′) is the gradient of pattern-perimeterdensity at location r′.

Reference is now made to FIGS. 4A and 4B, which illustrate a top viewand a cross-section view, respectively, of wafer 403 configured to beexposed to a plasma (e.g., plasma 208 of FIG. 2 ), consistent with theembodiments of the present disclosure. As shown in FIG. 4A, Die 410 maybe fabricated on wafer 403 using semiconductor fabrication andprocessing methods, or microelectromechanical systems (MEMS) fabricationtechniques. Wafer 403 may comprise a substrate for fabrication ofmicroelectronic components of an integrated chip, and may be made from asemiconducting material including, but not limited to, silicon (Si),germanium (Ge), gallium arsenide (GaAs), or the like. In someembodiments, wafer 403 may be made of a insulating material such assilicon dioxide (SiO₂), glass, ceramics, or the like.

Wafer 403 may comprise more than one die fabricated in a repeatingpattern along X- and Y-axes. In the context of this disclosure, a “die”may refer to a block of semiconducting material (e.g., wafer 403) onwhich a functional integrated circuit is fabricated. Die 410 maycomprise electronic components including, but not limited to,semiconducting devices such as Metal-Oxide-Semiconductor Field EffectTransistors (MOSFETs), capacitors, diodes, resistors, among otherdevices. In some embodiments, more than one die on wafer 403 may have asimilar die pattern as die 410. Die pattern, as used herein, refers tothe layout of devices and circuitry within a die. It is appreciated thatdies may have dissimilar patterns as well, based on the desired functionand application.

FIG. 4B illustrates a cross-section view of wafer 403 comprising die410, along cross-section 413 of FIG. 4A. Wafer 403 may be placed on astage (e.g., stage 204 of FIG. 2 ) in a plasma reactor (e.g., plasmaetch chamber 210 of FIG. 2 ) to perform plasma etching. In semiconductorfabrication and processing, plasmas are used, for example to “dry” etchfeatures with vertical sidewalls and high aspect ratios into materialssuch as silicon, silicon dioxide, or glass. In the context of thisdisclosure, aspect ratio refers to a ratio of depth to width of afeature. For example, aspect ratio of a trench 200 μm wide and 4 mm deepis 20. Plasma etching or dry etching, in comparison to wet etching, maybe desirable to obtain anisotropic etch profiles, among other things.

During plasma etching, wafer 403 may be subjected to a plasma having aplasma sheath region 421 formed at the interface between plasma and anelectrode, a chamber wall, or a sample (e.g., wafer 403). A plasmasheath region may be a dark, electron-depleted, positively-charged,boundary region containing positive ions and neutral species. The plasmasheath region is formed because electrons in a plasma are more mobile(higher temperature and lower mass) than ions, and consequently, escapefrom plasma at much faster speed than ions if there is no confiningpotential barrier. Positive charges in the plasma sheath region canprevent more positive ions from diffusing out of the plasma, and canalso create a potential barrier to prevent electrons from diffusing outof the plasma. A plasma sheath may also create a positive plasmapotential with respect to the grounded chamber walls, or groundedelectrode, or the sample. FIG. 4B shows a plasma potential profile 420including plasma sheath region 421, and ion trajectories 422, 423, and424 directed towards wafer 403.

As shown in FIG. 4B, ion trajectories 422 and 424 originating fromplasma sheath regions 421 may be different from ion trajectory 423originating from a uniform potential regions of the plasma. Iontrajectory 423 may be substantially perpendicular to a surface of wafer403, or a center die 410C fabricated on wafer 403. Ion trajectories 422and 424 may be incident on die 410 at a non-zero angle with respect tocenter axis 404. Because the etchant ions have different trajectories,the etch characteristics including, but not limited to, etch profile,etch rate, etch anisotropy, or etch asymmetry of features may bedifferent. In some embodiments, etch profile of a feature may be basedon the location of the die on wafer 403.

As an example, etch profiles 425 and 427 of a feature 408 of peripheraldies 410 may be asymmetrical in comparison to the etch profile 426 ofcenter die 410C. Etch profiles 425 and 427 illustrate after-etchinspection (AEI) profiles of feature 408 of peripheral dies 410 locatedat a radial distance r from a centerpoint (X=0, Y=0). Etch profile 426illustrates an AEI profile of feature 408 of center die 410C.

Reference is now made to FIG. 5 , which illustrates a schematic of a dielocated at a radial distance r from the center of the wafer, consistentwith embodiments of the present disclosure. Die 510 may comprise aperipheral die on a wafer (e.g., wafer 403 of FIG. 4A), and may besubstantially similar to peripheral die 410 of FIG. 4B. Die 510 may belocated at a radial distance r measured from the center of the wafer tocenter of die 510 having local coordinates (x₀, y₀). Die 510 may includean exemplary feature 508, analogous to feature 408 of FIG. 4A. Thelocation coordinates of feature 508 may be represented using linearcoordinates (x_(i), y_(i)) at a distance r′ from (x₀, y₀). It isappreciated that although die 510 is illustrated as including feature508, it may comprise a plurality of features similar or dissimilar tofeature 508.

Die 510 may comprise microelectronic devices including, but not limitedto, transistors, diodes, resistors, capacitors, and circuitry comprisingmicroelectronic devices arranged in a pattern or a layout. In someembodiments, the pattern or the layout may be predetermined based on theapplication. Pattern density of a die refers to the number of devices ina unit area of the die. Based on the arrangement of devices, die 510 maycomprise regions having a high pattern density (e.g., dense region 522),or regions having a lower pattern-perimeter density (e.g., isolatedregion 524). Although not explicitly illustrated in FIG. 5 , it isappreciated that there may be one or more regions with intermediate orvarying density levels within a die.

In some embodiments, information associated with pattern-perimeterdensity of a die or pattern-perimeter density variation across a die maybe represented in a corresponding pattern-perimeter density mapindicating the physical layout of devices. Information associated with apattern-perimeter density map of die 510, in combination withinformation associated with ion tilt, ion angle, ion energy, flux atwafer scale from wafer-scale model 310, may be used by die-scale model320 to simulate the characteristics of an ion, neutral or radicals inthe plasma. The ion characteristics derived from die-scale modeling mayinclude, but are not limited to, modification of ion tilt, modificationof ion energy, modification of flux of ions, modification of ion angulardistribution, modification of neutral flux, modification of radical fluxor the like, based on one or both of micro-loading effects and surfacecharging effects.

Reference is now made to FIG. 6 , which illustrates a deviation of iontrajectory at a junction between the dense and isolated regions of adie, consistent with embodiments of the present disclosure. Die 610 maycomprise a dense region 622, an isolated region 624, and a junctionregion 623 between the dense and the isolated regions. In someembodiments, ion 605 in a plasma may be directed toward die 610 along aninitial path or an initial ion trajectory 610 a. Ion 605 or an ion beam(not shown) comprising a plurality of ions may be incident on feature608 located in isolated region 624. As ion 605 approaches the surface offeature 608, differential surface charging effects may cause a deviationof ion trajectory from initial ion trajectory 610 a to final iontrajectory 620 a “close” to the incident surface. The deviation of iontrajectory may occur or initiate at a distance d from the incidentsurface, and may be in a range of 2 mm or less, 1 mm or less, 500 μm orless, 200 μm or less, or 100 μm or less from the incident surface.

Reference is now made to FIGS. 7A-7C, which illustrate the schematics ofa high pattern-perimeter density and a low pattern-perimeter densityregions comprising a feature and differential surface charging effects,consistent with embodiments of the present disclosure.

FIG. 7A illustrates a schematic of a dense region (highpattern-perimeter density) 722, analogous to dense regions 522 and 622of FIGS. 5 and 6 , respectively, and a schematic of an isolated region(lower pattern-perimeter density) 724, analogous to isolated regions 524and 624 of FIGS. 5 and 6 , respectively. Dense region 722 may comprise aplurality of features 708 arranged in a repeating manner, for example, amatrix, an array, a pattern, or randomly arranged. Isolated region 724may comprise fewer features in comparison to dense region 722. Feature708 may include, but is not limited to, an alignment mark, a trench, ametal contact pad, a transistor gate, a via, or other such feature.

FIG. 7B illustrates a plot of simulated differential surface chargingeffect for dense and isolated regions using a die-scale model,consistent with embodiments of the present disclosure. As shown in FIG.7B, isolated region 724 is at a higher potential in comparison to denseregion 722 at any given distance above the incident surface. The surfaceand sidewalls of dense region 722 are mostly charged negative by theelectrons while most of the positively charged ions escape into thetrench since the latter has more anisotropic angular distribution. Theisolated region 724 receives equal current of electrons and ionsaveraged over an RF cycle. This differential charging of the dense andisolated surface may generate a surface potential measured in Volts (V),represented on the Y-axis of the plot.

FIG. 7C illustrates a plot of simulated differential surface chargingeffect for dense and isolated regions using a die-scale model,consistent with embodiments of the present disclosure. As shown in FIG.7C, the surface potential of isolated region 724 is higher in comparisonto dense region 722 at any given point across the incident surface. Thesurface and sidewalls of dense region 722 are mostly charged negative bythe electrons while most of the positively charged ions escape into thetrench since the latter has more anisotropic angular distribution. Theisolated region 724 receives equal current of electrons and ionsaveraged over an RF cycle. This differential charging of the dense andisolated surface may generate a surface potential measured in Volts (V),represented on the Y-axis of the plot.

Reference is now made to FIGS. 8A and 8B, which illustrate theschematics of high pattern-perimeter density and low pattern-perimeterdensity regions, and the ion angular distribution in a highpattern-perimeter density region, consistent with embodiments of thepresent disclosure.

FIG. 8A illustrates a schematic of a region of a die comprising a highpattern-perimeter density region 822, a low pattern-perimeter densityregion 824, and a junction region 823. High pattern-perimeter densityregion 822 may comprise one or more features 808. Low pattern-perimeterdensity region 824 may comprise none or fewer features in comparison tohigh density region 822.

FIG. 8B illustrates the simulated modified ion angular distribution ofincident ions on the surface of the die at junction region 823. The ionangular distributions may be represented as a normal, a standard, or aGaussian distribution function. FIG. 8B illustrates the modified ionangular distribution of ions incident on the surface of the die atjunction region 823, and the mean (or the median or the mode) of the ionangle may be modified by a finite positive offset. A die-scale model(e.g., die-scale model 320 of FIG. 3 ) may be configured to simulate themodification of ion angular distribution based on pattern-perimeterdensity. In some embodiments, die-scale model 320 may be furtherconfigured to simulate modification of more than one characteristic ofthe particles of the plasma including, but not limited to, ion tilt, ionangular distribution, ion trajectory, ion flux, ion energy,

Reference is now made to FIG. 9 , which illustrates a plasma simulationsystem 900 comprising a data-driven model for simulating a plasma etchprocess, consistent with embodiments of the present disclosure. Plasmasimulation system 900 may include a wafer-scale model 910, a die-scalemodel 920, training image 930, a machine learning network 940, trainedimage 950. It is appreciated that simulation system 900 may compriseother relevant components (not illustrated) as well.

Wafer-scale model 910 may be substantially similar to and may performsubstantially similar functions as wafer-scale model 310 of FIG. 3 .Wafer-scale model 910, analogous to wafer-scale model 310, may comprisea data input module, a data processing module, and a post-processmodule. Wafer-scale model 910 may be configured to predict a plasmacharacteristic including, but not limited to, ion tilt, ion trajectory,ion angular distribution, or ion flux, based on the information obtainedfrom data input module such as process conditions for etch, chambergeometry, etc. Wafer-scale model 910 may be further configured topredict, for example, ion tilt at a given location on a wafer (e.g.,wafer 403 of FIG. 4 ) located at a radial distance r from the center ofthe wafer. In some embodiments, information associated with thepredicted plasma characteristic using wafer-scale model 910 may bestored in a storage module (not shown) of machine learning network 940.

Die-scale model 920 may be substantially similar to and may performsubstantially similar functions as die-scale model 320 of FIG. 3 .Die-scale model 920, analogous to die-scale model 320, may comprise adata input module, and one or more sub-modules for modeling themicro-loading effects and the differential surface charging effectsbased on a pattern-perimeter density, or pattern-perimeter densityvariation, or pattern layout of the die. Die-scale model 920 may beconfigured to predict a modification of the plasma characteristicpredicted from wafer-scale model 920 based on the pattern-perimeterdensity map, or pattern-perimeter density variation map. In someembodiments, die-scale model 920 may be configured to store informationassociated with the predicted modification of the plasma characteristicin a storage module of machine learning network 940.

In some embodiments, plasma simulation system 900 may be an automatedmachine learning network trained to receive or extract training image930 from a database, unprompted. Training image 930 may be anafter-develop image of a feature, or an after-develop image of a regionof interest on a wafer (e.g., wafer 403 of FIG. 4 ), or may include aplurality of after-develop images of a feature. Training image 930 maybe acquired using an image acquirer of an inspection system. Afterreceiving or acquiring training image 930 or information associated withtraining images 930, machine learning network 940 may extract relevanttrained features, unprompted. The extracted trained features may bestored in a storage module (not shown) or temporarily stored in arepository (not shown). The storage module may be accessed by machinelearning network 940, or a user of plasma simulation system 900.

In some embodiments, machine learning network 940 may be configured toextract feature information from training image 930. Machine learningnetwork 320 may also extract relevant features from information filecomprising GDS format files or OASIS format files. Machine learningnetwork 940 may include, for example, an artificial intelligence system,a neural network, a convolutional encoder-decoder, or a deep-learningtechnique, a software implemented algorithm, or the like. The featureextraction architecture of machine learning network 940 may comprise aconvolutional neural network, for example. In some embodiments, a linearclassifier network of deep learning architecture may be adopted as astarting point to train and build feature extraction architecture ofmachine learning network 940.

In some embodiments, machine learning network 940 may include apattern-perimeter extractor (not shown) configured to extractinformation or pattern-perimeters from training image 930.Pattern-perimeter extractor may be a mathematical algorithm, asoftware-implemented algorithm, image processing algorithm, or the like.Pattern-perimeter extractor may be integrated into an image acquirer(not shown) or may be configured to operate as a separate, stand-aloneunit configured to process training image 930. In some embodiments,pattern-perimeter extractor may comprise an image processing unit (notshown) configured to adjust brightness, contrast, saturation, flatness,noise filtering, etc. of training image 930 prior to storage in storagemodule of machine learning network 940.

In some embodiments, machine learning network 940 may further comprisean image acquirer, image enhancer, display device, or the like. Machinelearning network 940 may be configured to extract the pattern-perimeterinformation from training image 930 and to receive informationassociated with predicted plasma characteristics from wafer-scale model910 and die-scale model 920.

In some embodiments, machine learning network 940 may be furtherconfigured to generate trained image 950 based on the information fromtraining image 930, wafer-scale model 910, and die-scale model 920.Trained image 950 may comprise a simulated after-etch image based on thepredicted etch profile of a feature using wafer-scale model 910 anddie-scale model 920. Trained image 950, representing a simulated etchcharacteristic of a feature after stripping the photoresist, may bereviewed by multiple reviewers or users requesting the information. Insome embodiments, trained image 950 may be retrieved by a user prompt ata later time for review and in-depth analysis. Trained image 950 may bestored in a suitable format, for example, a Joint Photographic ExpertsGroup (JPEG) file, a Portable Network Graphic (PNG) file, a PortableDocument Format (PDF) file, a Tagged Image File Format (TIFF) file, orthe like.

FIG. 10 is a process flow chart illustrating an exemplary simulationmethod 1000 for simulating a plasma etch process, consistent withembodiments of the present disclosure. The simulation method may beperformed using a physical etch model (e.g., multi-scale plasma etchmodel 300 of FIG. 3 ) in a plasma process simulation system (e.g.,plasma process simulation system 100 of FIG. 1 ). For example, aprocessor (e.g., processor 180 of FIG. 1 ) may include the modelingprogram or algorithm and may be programmed to implement the simulationmethod. It is appreciated that steps performed in simulation method 1000may be reordered, added, removed, or edited, as appropriate.

In step 1010, the simulation method may include predicting a firstcharacteristic of a particle of a plasma in a first scale based on afirst plurality of parameters. Plasma may be a neutral ionized gascomprising positively charged ions (etchant species), free radicals,neutral species, or electrons. The first characteristic may comprise aphysical characteristic of the particle such as ion tilt, ion energy,ion trajectory, ion flux, ion angular distribution, among othercharacteristics. The first scale may be a wafer-scale having dimensionsup to 400 mm. The first plurality of parameters may comprise chambergeometry, process conditions, or a wafer location. A wafer-scale model(e.g., wafer-scale model 310 of FIG. 3 ) may be used to predict thefirst characteristic of the plasma in wafer-scale.

In some embodiments, predicting a plasma characteristic such as iontilt, for example, may include predicting a plasma sheath profile inwafer-scale using the wafer-scale model. A data processing module (e.g.,data processing module 305 of FIG. 3 ) of wafer-scale model 310 may beconfigured to receive information associated with the plasma processingchamber and process conditions. The data processing module may befurther configured to predict a profile of plasma potential in theplasma sheath region based on the plasma processing chamber geometry andprocess conditions. Predicting the ion tilt at a location on the waferat a radial distance r may further include tracking the trajectory ofthe ion in the presence of electric potential gradient. The wafer-scalemodel may also be configured to predict plasma characteristics such asion angle distribution, ion energy, ion flux, ion trajectory, or thelike.

In step 1020, the simulation method may include predicting a secondcharacteristic of the particle in a second scale based on a modificationof the first characteristic caused by a second plurality of parameters.The second scale may include a die-scale having dimensions in the rangeof 5 mm-20 mm or more. A die (e.g., peripheral die 410 of FIG. 4A) mayinclude a plurality of features having dimensions in the range of 5nm-100 μm or more. A die-scale model (e.g., die-scale model 320 of FIG.3 ) may be configured to receive one or more of the first characteristicof the plasma and predict one or more of the second characteristics indie-scale.

The second characteristic may include a modification of ion tilt, ionenergy, ion trajectory, ion flux, ion angular distribution, neutralflux, radical flux, among other characteristics. In some embodiments,the second characteristic of the particles may comprise a physicalcharacteristic of the plasma particles in one or more scales such aswafer-scale, die-scale, or feature-scale. The modification ofcharacteristics of an ion in a plasma approaching a surface of a featureon a die may be caused by factors including, but not limited to,pattern-perimeter density, pattern-perimeter density variation, orpattern-perimeter layout, among other factors. For example, differentialsurface charging effects due to pattern-perimeter density variation maymodify the ion tilt, ion trajectory, ion angular distribution, ionenergy, or ion flux of the incident ions. As a different example,micro-loading effects due to pattern-perimeter density variation maymodify the flux of neutrals or radicals incident on the die surface. Thedie-scale model may be configured to predict the modification of plasmacharacteristics from wafer-scale based on a location r′ of the featurewith respect to the center of the die (x₀, y₀).

In some embodiments, predicting the modification of plasmacharacteristics as the ion approaches the die surface may includeaccessing a pattern-perimeter layout of the die. The pattern-perimeterlayout may include a pattern-perimeter density map or apattern-perimeter density variation map. The pattern-perimeter densitymap may comprise regions of high pattern-perimeter density (denseregions) and lower pattern-perimeter density (isolated regions) based onthe die design for a desired application. The die-scale model maypredict the modification of plasma characteristics based on thepattern-perimeter density map or the pattern-perimeter density variationmap.

In the die-scale model, predicting the modification of plasmacharacteristics based on the pattern-perimeter density map or thepattern-perimeter density variation map may include identifying a firstregion of the die having a first pattern-perimeter density and a secondregion of the die having a second pattern-perimeter density differentfrom the first pattern-perimeter density. The first and the secondregions may include the dense and the isolated regions, respectively. Itis appreciated that the first and the second regions may include theisolated and the dense regions, respectively, as well.

Upon accessing the pattern-perimeter density map, the die-scale modelmay predict a concentration gradient of etchants (neutrals or radicals)based on the identified dense and the isolated regions. Theconcentration gradient, caused by the higher consumption of etchants indense regions compared to isolated regions, may influence diffusion ofthe etchants from isolated to dense regions on the die surface. Thedie-scale model may predict the modification of plasma characteristicsat a location r′ based on the neighboring patterns, pattern-perimeterdensity, or pattern-perimeter density variation.

In step 1030, the simulation method may include simulating an etchcharacteristic of a feature based on the first and the secondcharacteristics of the charged particle predicted from the wafer-scaleand the die-scale model. In some embodiments, simulation method mayfurther include simulating the etch characteristic based on one or morephysical characteristics of the plasma particles, or based on a physicalcharacteristic in one or more scales. The multi-scale plasma etch modelcomprising the wafer-scale and the die-scale model may be configured tosimulate an etch characteristic including, but not limited to, etchprofile, etch rate, etch uniformity, etch selectivity, or the like, of afeature on the die. In some embodiments, the simulation method maygenerate an image of the simulated etch characteristic, or generateinformation associated with the simulated etch characteristic in agraphic format or a tabulated format, or other formats.

Reference is now made to FIG. 11 , which illustrates a process flowchartillustrating an exemplary simulation method 1100 for simulating a plasmaetch process using a data-driven model, consistent with embodiments ofthe present disclosure. The simulation method may be performed using adata-driven etch model (e.g., data-driven etch model 334 of FIG. 3 ) ina plasma process simulation system (e.g., plasma process simulationsystem 100 of FIG. 1 ). For example, a processor (e.g., processor 180 ofFIG. 1 ) may include the modeling program or algorithm and may beprogrammed to implement the simulation method. It is appreciated thatsteps performed in simulation method 1100 may be reordered, added,removed, or edited, as appropriate.

In step 1110, the simulation method may include acquiring a first imageof the feature. The first image may comprise a training image (e.g.,training image 930 of FIG. 9 ) an after-develop image of the feature, ora plurality of after-develop images. The image(s) may be acquired,retrieved, accessed, or obtained from a database, a storage module, orin some cases from an image acquirer of an optical inspection system inreal-time.

In step 1120, the simulation method may include identifying the featurebased on a pattern-perimeter information from the acquired trainingimage. The pattern-perimeter information may be extracted and thefeature may be identified using a pattern-perimeter extractor. Thepattern-perimeter information may comprise global structuralinformation, for example, reference fiducials for a photolithographyprocess on the wafer, alignment marks, reference features on a wafer,features to be etched, etc. Identification of the feature may beperformed by a feature extraction algorithm, for example.

In step 1130, the simulation method may include predicting an etchprofile of the feature to be etched using a plasma etch process.Predicting the etch profile may include predicting a firstcharacteristic of a particle of a plasma in a first scale based on afirst plurality of parameters and predicting a second characteristic ofthe particle in a second scale based on a modification of the firstcharacteristic caused by a second plurality of parameters.

The first characteristic may comprise ion tilt, ion energy, iontrajectory, ion flux, ion angular distribution, among othercharacteristics. The first scale may be a wafer-scale having dimensionsup to 400 mm. The first plurality of parameters may comprise chambergeometry, process conditions, or a wafer location. A wafer-scale model(e.g., wafer-scale model 310 of FIG. 3 ) may be used to predict thefirst characteristic of the plasma in wafer-scale.

The second scale may include a die-scale having dimensions in the rangeof 5 mm-20 mm or more. A die (e.g., peripheral die 410 of FIG. 4A) mayinclude a plurality of features having dimensions in the range of 5nm-100 μm or more. A die-scale model (e.g., die-scale model 320 of FIG.3 ) may be configured to receive one or more of the first characteristicof the plasma and predict one or more of the second characteristics indie-scale. The second characteristic may include a modification of iontilt, ion energy, ion trajectory, ion flux, ion angular distribution,neutral flux, radical flux, among other characteristics. Themodification of characteristics of an ion, neutral or radical in aplasma approaching a surface of a feature on a die may be caused byfactors including, but not limited to, pattern-perimeter density,pattern-perimeter density variation, or pattern layout, among otherfactors. For example, differential surface charging effects due topattern-perimeter density variation may modify the ion tilt, iontrajectory, ion angular distribution, ion energy, or ion flux of theincident ions. As a different example, micro-loading effects due topattern-perimeter density variation may modify the flux of neutrals orradicals incident on the die surface. The die-scale model may beconfigured to predict the modification of plasma characteristics fromwafer-scale based on a location r′ of the feature with respect to thecenter of the die (x₀, y₀).

In some embodiments, predicting the modification of plasmacharacteristics as the ion approaches the die surface may includeaccessing a pattern layout of the die. The pattern layout may include apattern-perimeter density map or a pattern-perimeter density variationmap. The pattern-perimeter density map may comprise regions of highpattern-perimeter density (dense regions) and lower pattern-perimeterdensity (isolated regions) based on the die design for a desiredapplication. The die-scale model may predict the modification of plasmacharacteristics based on the pattern-perimeter density map or thepattern-perimeter density variation map.

In the die-scale model, predicting the modification of plasmacharacteristics based on the pattern-perimeter density map or thepattern-perimeter density variation map may include identifying a firstregion of the die having a first pattern-perimeter density and a secondregion of the die having a second pattern-perimeter density differentfrom the first pattern-perimeter density. The first and the secondregions may include the dense and the isolated regions, respectively. Itis appreciated that the first and the second regions may include theisolated and the dense regions, respectively, as well.

After accessing the pattern-perimeter density map, the die-scale modelmay predict a concentration gradient of etchants based on the identifieddense and the isolated regions. The concentration gradient of etchantscaused by the higher consumption of etchants in dense regions comparedto isolated regions may result in diffusion of uncharged particles ofthe plasma including neutrals or radicals from isolated to dense regionson the die surface. The die-scale model may predict the modification ofplasma characteristics at a location r′ based on the neighboringpatterns, pattern-perimeter density, pattern-perimeter densityvariation, or a pattern-perimeter density gradient.

In step 1140, the simulation method may include generating a secondimage comprising the predicted etch profile of the feature. The secondimage may comprise a trained image (e.g., trained image 950 of FIG. 9 )of a simulated after-etch profile of the feature. The simulated imagemay be generated based on predicted plasma characteristics andmodification of plasma characteristics in wafer-scale and die-scale,respectively. The simulated image may be a predicted etch profile of adeveloped feature in the training image.

Reference is now made to FIG. 12 , which illustrates an exemplarymulti-scale physical etch model 1200, consistent with embodiments of thepresent disclosure. Multi-scale etch model 1200 may comprise a fab-scalemodel 1210, a chamber-scale model 1220, a wafer-scale model 1230, adie-scale model 1240, and a feature-scale model 1250, to generate asimulated etch profile based on the multi-scale physical etch modelapproach.

A semiconductor wafer processing facility is generally referred to as afabrication facility, or a “fab.” A fab may be equipped with one or moreplasma reactors configured to perform plasma processes including plasmadeposition, plasma etching, plasma treatments, or the like. To increasethe wafer throughput, reduce the cycle time, and improve processcontrol, multiple plasma reactors may be configured and utilized toperform a single process, a single step, or a part of a process of thewafer processing cycle. For example, multiple plasma reactors may beengaged to perform a polysilicon gate etch on a large batch of wafers,and the etch characteristics of the features produced on all wafers fromevery reactor are expected to be substantially similar or within thespecification. However, in practice, reactors do not process wafersidentically and therefore, cause variations in the etch characteristicsof features produced. In some instances, features of wafers processed inthe same tool may vary between cycles or even within the same cycle.These variations in etch characteristics may be caused by factorsincluding, but not limited to, chamber processing history, chambercharacteristics, hardware settings, maintenance schedules, chamber age,etch chemistries, etc. The variability in etch performance and etchcharacteristics of features may negatively impact the process yields,throughput, costs, and in some cases may also cause device failure.

One of the several methodologies to produce consistent etch featuresacross multiple wafers processed in multiple chambers includes “chambermatching.” Chamber matching may include, among other things, optimizingthe operating parameters on a reference “golden” chamber and duplicatingthe optimized parameters to multiple chambers in the facility, oroptimizing the operating parameters of every chamber within thespecification and tolerance levels. The accuracy of modeling etchcharacteristics of a feature within a die on a wafer processed in aplasma chamber may be enhanced using a multi-scale physical etch modelsuch as multi-scale physical etch model 1200, for example.

In some embodiments, multi-scale physical etch model 1200 may comprisetwo or more models. Although FIG. 12 shows five models, there may bemore or fewer models, and the models may include one or moresub-modules. Information obtained from one model may be shared betweenone or more models, as indicated by the dotted lines in FIG. 12 . Insome embodiments, fab-scale model 1210 may be configured to predictboundary conditions of chambers within the fab, based on factorsincluding, but not limited to, chamber conditions, chamber processinghistory, chamber limitations, or chamber maintenance schedules. Forexample, a chamber X in the fab may not be qualified to perform an etchprocess utilizing silane. Fab-scale model 1210 may obtain informationassociated with chamber X and may exclude the chamber from the chambermatching boundary conditions. Fab-scale model 1210 may further beconfigured to perform chamber matching based on the information obtainedfrom one or more models such as chamber-scale model 1220, wafer-scalemodel 1230, die-scale model 1240, or feature-scale model 1250.

In some embodiments, boundary conditions may be applied to two or morechambers configured to perform a plasma process. For example, aplurality of chambers within the boundary conditions may form a networkthrough which the information associated with one or more chambers maybe shared. The information may be shared within the same scale or acrossmultiple scales including chamber scale, wafer scale, die scale, orfeature scale.

Chamber-scale model 1220 may be configured to predict characteristics ofa chamber. The characteristics of the chamber may include, but are notlimited to, chamber wall conditions, gas inlet location, pump outletlocation, chamber geometry, chamber material, or the like. The boundaryconditions for the chamber-scale model 1220 may be determined based on,for example, the locations within the chamber from which the informationabout the desired characteristic(s) may be extracted. In someembodiments, chamber-scale model may comprise a 1-dimensional (1D)network model.

Wafer-scale model 1230, die-scale model 1240, and feature-scale model1250 may be analogous to wafer-scale model 310, die-scale model 320, andfeature-scale model 330 of FIG. 3 , respectively, and may performsubstantially similar functions. In some embodiments, the boundaryconditions, physical models, and algorithms used may be different basedon the scale or the level. In some embodiments, the boundary conditions,physical models, and algorithms at one or more levels may be influencedby the boundary conditions, physical models, and algorithms at otherlevels. For example, the boundary conditions for fab-scale model 1210,though different from boundary conditions of chamber-scale model 1220,may be interdependent.

Using multi-scale physical etch model 1200, the etch characteristics ofa feature on a wafer (e.g., wafer 403 of FIG. 4A) may be predicted basedon information obtained from one or more scales including die-scale,wafer-scale, chamber-scale, or fab-scale. As an example, the etchcharacteristics of a feature may vary based on the pattern-perimeterdensity map in the die-scale, and the etch characteristics may bepredicted based on at least information obtained using die-scale model320. In some embodiments, the etch characteristics of a feature may varybased on the pattern-perimeter density map and the ion trajectory or theion tilt at a location on the wafer. The etch characteristics of thefeature may be predicted based on the information obtained fromdie-scale model 320 and wafer-scale model 310.

In some embodiments, the etch characteristics of a feature, in additionto die-scale and wafer-scale factors, may vary based on the chamber wallconditions, for example. The information associated with acharacteristic of the chamber may be obtained using chamber-scale model1220. The etch recipe, etch process, etch duration, etch chemistry, orthe like, of the etch process performed may influence the characteristicof the chamber such as chamber wall condition, chamber pressure, gasinlet locations, pump outlet locations. The end-of-line yield of a wafermay be influenced by one or more chamber characteristics. For example,the etch characteristics of a feature on a wafer processed in a chamberconfigured to run lean etch chemistry may be different from the chamberconfigured to run polymerization etch chemistry. Therefore, informationobtained from chamber-scale model 1220 may influence the information inwafer-scale model 1230, or die-scale model 1240, or feature-scale model1250.

In some embodiments, the etch characteristics of a feature, in additionto die-scale, wafer-scale, and chamber-scale, may vary based on thechamber in which the etch process is performed, for example. Theinformation associated with the chamber of the plurality of chambers inthe fab may be obtained using fab-scale model 1210. In some embodiments,information obtained from feature-scale model 1250 may be used toperform chamber matching in fab-scale model 1210 by adjusting theoperating parameters, or fine tuning the etch conditions.

In some embodiments, multi-scale physical etch model 1200 may beconfigured to control etch tilt, as referred to in FIG. 4B. The etchtilt may be adjusted, for example, by adjusting a characteristic of afocus ring (e.g., focus ring 205 of FIG. 2 ). The characteristic offocus ring 205 may include a horizontal position, a vertical position, ashape, a construction material, a cross-section, or an applied voltageto the focus ring. For example, the voltage applied to the focus ringmay be adjusted to adjust the trajectory of the charged particles of theplasma toward the wafer edge, or the vertical position of the focus ringmay be physically adjusted to adjust the etchant confinement at the edgeof the wafer, shape of the plasma sheath, or the trajectory of the ionsof the plasma immediately above the wafer surface. The modifiedtrajectory or the ion tilt approaching the wafer surface may influencethe etch profile of the feature.

A non-transitory computer readable medium may be provided that storesinstructions for a processor of a controller (e.g., controller 50 ofFIG. 1 ) to carry out image inspection, image acquisition, imageprocessing, database management, numerical analysis of data, executingmodeling and simulation algorithms, data storage, generating images,etc. Common forms of non-transitory media include, for example, a floppydisk, a flexible disk, hard disk, solid state drive, magnetic tape, orany other magnetic data storage medium, a Compact Disc Read Only Memory(CD-ROM), any other optical data storage medium, any physical mediumwith patterns of holes, a Random Access Memory (RAM), a ProgrammableRead Only Memory (PROM), and Erasable Programmable Read Only Memory(EPROM), a FLASH-EPROM or any other flash memory, Non-Volatile RandomAccess Memory (NVRAM), a cache, a register, any other memory chip orcartridge, and networked versions of the same.

Embodiments of the present disclosure can be further described by thefollowing clauses.

1. A method for simulating a plasma etch process, the method comprising:

-   -   predicting a first characteristic of a particle of a plasma in a        first scale based on a first plurality of parameters;    -   predicting a second characteristic of the particle in a second        scale based on a modification of the first characteristic caused        by a second plurality of parameters; and    -   simulating an etch characteristic of a feature based on the        first and the second characteristics of the particle.

2. The method of clause 1, further comprising predicting a sheathprofile of the plasma in the first scale based on the first plurality ofparameters, wherein the first scale comprises a wafer-scale.

3. The method of clause 2, wherein predicting the first characteristiccomprises determining a gradient of the predicted sheath profile, andwherein the first characteristic comprises an angle of incidence, atrajectory, or an energy of the particle directed towards a wafer.

4. The method of clause 3, wherein the first plurality of parameterscomprises geometry of a plasma reactor configured to perform the plasmaetch process, a process condition for the plasma etch process, or alocation on the wafer.

5. The method of any one of clauses 3 and 4, wherein predicting thesecond characteristic comprises predicting a modification of the angleof incidence, the trajectory, or the energy of the particle in thesecond scale, and wherein the second scale comprises a die-scale.

6. The method of clause 5, wherein predicting the second characteristicfurther comprises:

-   -   accessing a layout of a die, the layout comprising a        pattern-perimeter density map; and    -   predicting the second characteristic of the particle based on        the pattern-perimeter density map, wherein the particle        comprises a charged particle or an uncharged particle.

7. The method of clause 6, wherein predicting the second characteristicof the charged particle of the plasma comprises:

-   -   identifying, based on the pattern-perimeter density map, a first        region of the die having a first pattern-perimeter density and a        second region of the die having a second pattern-perimeter        density different from the first pattern-perimeter density;    -   predicting an electric potential gradient between the identified        first and the second regions;    -   and predicting the second characteristic of the charged particle        based on the electric potential gradient.

8. The method of clause 7, wherein predicting the electric potentialgradient comprises:

-   -   predicting a first electric potential of the first region        comprising a dense region having a high pattern-perimeter        density; and    -   predicting a second electric potential of the second region        comprising an isolated region having a lower pattern-perimeter        density, wherein the first and the second electric potentials        are different.

9. The method of clause 8, wherein the first electric potential is lowerthan the second electric potential.

10. The method of any one of clauses 7-9, wherein predicting the secondcharacteristic of the uncharged particle of the plasma comprises:

-   -   predicting a concentration gradient of an etchant between the        identified first and the second regions;    -   predicting a diffusion flux of the etchant based on the        concentration gradient; and    -   predicting the second characteristic of the uncharged particle        based on the diffusion flux.

11. The method of clause 10, wherein predicting the concentrationgradient comprises:

-   -   predicting a first concentration of the etchant in the first        region; and    -   predicting a second concentration of the etchant in the second        region, wherein the first and the second concentrations are        different.

12. The method of clause 11, wherein the first concentration is lowerthan the second concentration.

13. The method of any one of clauses 6-12, further comprising predictingthe second characteristic of the particle in the second scale based on apattern-perimeter density gradient of the die and a Gaussian kernel,wherein the Gaussian kernel is a multi-length scale kernel comprising alength scale ranging from 5 nm to 50 μm.

14. The method of clause 13, wherein the second plurality of parameterscomprises the layout, a pattern-perimeter density, a pattern-perimeterdensity variation, or the pattern-perimeter density gradient of the die.

15. The method of any one of clauses 6-14, wherein simulating the etchcharacteristic comprises simulating an etch rate, an etch profile, or anetch asymmetry of the feature based on the pattern-perimeter density mapof the die.

16. A method for generating a simulated image of a feature, the methodcomprising:

-   -   acquiring a first image of the feature;    -   identifying the feature based on a pattern-perimeter information        from the image;    -   predicting an etch profile of the feature to be etched using a        plasma etch process, the predicting comprising:    -   predicting a first characteristic of a particle of a plasma in a        first scale based on a first plurality of parameters; and    -   predicting a second characteristic of the particle in a second        scale based on a modification of the first characteristic caused        by a second plurality of parameters; and    -   generating a second image comprising the predicted etch profile        of the feature.

17. The method of clause 16, further comprising acquiring the firstimage from a user-defined database, wherein the user-defined databasecomprises a graphic database system.

18. The method of any one of clauses 16 and 17, wherein identifying thefeature comprises comparing the pattern-perimeter information and atrained feature from a trained image, using a machine learning network.

19. The method of any one of clauses 16-18, wherein the first imagecomprises an after-develop image of the feature.

20. The method of any one of clauses 16-19, wherein the first scalecomprises a wafer-scale and the second scale comprises a die-scale.

21. The method of clause 20, wherein predicting the etch profile furthercomprises predicting a sheath profile of the plasma in the wafer-scalebased on the first plurality of parameters.

22. The method of clause 21, wherein predicting the first characteristiccomprises determining a gradient of the predicted sheath profile, andwherein the first characteristic comprises an angle of incidence, atrajectory, or an energy of the particle directed towards a wafer.

23. The method of clause 22, wherein the first plurality of parameterscomprises geometry of a plasma reactor configured to perform the plasmaetch process, a process condition for the plasma etch process, or alocation on the wafer.

24. The method of any one of clauses 22 and 23, wherein predicting thesecond characteristic comprises predicting a modification of the angleof incidence, the trajectory, or the energy of the particle in thedie-scale.

25. The method of clause 24, wherein predicting the secondcharacteristic further comprises:

-   -   accessing a layout of a die, the layout comprising a        pattern-perimeter density map; and    -   predicting the second characteristic of the particle based on        the pattern-perimeter density map, wherein the particle        comprises a charged particle or an uncharged particle.

26. The method of clause 25, wherein predicting the secondcharacteristic of the charged particle of the plasma further comprises:

-   -   identifying, based on the pattern-perimeter density map, a first        region of the die having a first pattern-perimeter density and a        second region of the die having a second pattern-perimeter        density different from the first pattern-perimeter density;    -   predicting an electric potential gradient based on the        identified first and the second regions; and    -   predicting the second characteristic of the charged particle        based on the electric potential gradient.

27. The method of clause 26, wherein predicting the electric potentialgradient comprises:

-   -   predicting a first electric potential of the first region        comprising a dense region having a high pattern-perimeter        density; and    -   predicting a second electric potential of the second region        comprising an isolated region having a lower pattern-perimeter        density, wherein the first and the second electric potentials        are different.

28. The method of clause 27, wherein the first electric potential islower than the second electric potential.

29. The method of any one of clauses 26-28, wherein predicting thesecond characteristic of the uncharged particle of the plasma comprises:

-   -   predicting a concentration gradient of an etchant between the        identified first and the second regions;    -   predicting a diffusion flux of the etchant based on the        concentration gradient; and    -   predicting the second characteristic of the uncharged particle        based on the diffusion flux.

30. The method of clause 29, wherein predicting the concentrationgradient comprises:

-   -   predicting a first concentration of the etchant in the first        region; and    -   predicting a second concentration of the etchant in the second        region, wherein the first and the second concentrations are        different.

31. The method of clause 30, wherein the first concentration is lowerthan the second concentration.

32. The method of any one of clauses 24-31, further comprisingpredicting the second characteristic of the particle in the second scalebased on a pattern-perimeter density gradient of the die and a Gaussiankernel, wherein the Gaussian kernel is a multi-length scale kernelcomprising a length scale ranging from 5 nm to 50 μm.

33. The method of clause 32, wherein the second plurality of parameterscomprises the layout, a pattern-perimeter density, a pattern-perimeterdensity variation, or the pattern-perimeter density gradient of the die.

34. A plasma etch simulation system, comprising:

-   -   a memory storing a set of instructions; and    -   a processor configured to execute the set of instructions to        cause the plasma etch simulation system to:        -   predict a first characteristic of a particle of a plasma in            a first scale based on a first plurality of parameters;        -   predict a second characteristic of the particle in a second            scale based on a modification of the first characteristic            caused by a second plurality of parameters; and        -   simulate an etch characteristic of a feature based on the            first and the second characteristics of the particle.

35. The system of clause 34, wherein the processor is configured toexecute the set of instructions to further cause the plasma etchsimulation system to:

-   -   predict a sheath profile of the plasma in the first scale based        on the first plurality of parameters;    -   determine a gradient of the predicted sheath profile; and    -   determine an angle of incidence, a trajectory, or an energy of        the particle directed towards a wafer based on the gradient of        the predicted sheath profile.

36. The system of clause 35, wherein the processor is configured toexecute the set of instructions to further cause the plasma etchsimulation system to:

-   -   access a layout of a die, the layout comprising a        pattern-perimeter density map; and    -   predict the second characteristic of the particle based on the        pattern-perimeter density map, wherein the particle comprises a        charged particle or an uncharged particle, and wherein the        second characteristic comprises a modification of the angle of        incidence, the trajectory, or the energy of the particle.

37. The system of clause 36, wherein the processor is configured toexecute the set of instructions to further cause the plasma etchsimulation system to:

-   -   identify, based on the pattern-perimeter density map, a first        region of the die having a first pattern-perimeter density and a        second region of the die having a second pattern-perimeter        density different from the first pattern-perimeter density;    -   predict an electric potential gradient based on the identified        first and the second regions; and    -   predict the second characteristic of the charged particle based        on the electric potential gradient.

38. The system of clause 37, wherein the processor is configured toexecute the set of instructions to further cause the plasma etchsimulation system to:

-   -   predict a first electric potential of the first region        comprising a dense region having a high pattern-perimeter        density; and    -   predict a second electric potential of the second region        comprising an isolated region having a lower pattern-perimeter        density, wherein the first and the second electric potentials        are different.

39. The system of any one of clauses 36-38, wherein the processor isconfigured to execute the set of instructions to further cause theplasma etch simulation system to:

-   -   predict a concentration gradient of an etchant between the        identified first and the second regions;    -   predict a diffusion flux of the etchant based on the        concentration gradient; and    -   predict the second characteristic of the uncharged particle        based on the diffusion flux.

40. The system of clause 39, wherein the processor is configured toexecute the set of instructions to further cause the plasma etchsimulation system to:

-   -   predict a first concentration of the etchant in the first        region; and    -   predict a second concentration of the etchant in the second        region, wherein the first and the second concentrations are        different.

41. The system of any one of clauses 34-40, wherein the processor isconfigured to execute the set of instructions to further cause theplasma etch simulation system to predict the second characteristic ofthe particle in the second scale based on a pattern-perimeter densitygradient of the die and a Gaussian kernel, wherein the Gaussian kernelis a multi-length scale kernel comprising a length scale ranging from 5nm to 50 μm.

42. A non-transitory computer readable medium storing a set ofinstructions that is executable by one or more processors of anapparatus to cause the apparatus to perform a method of simulating aplasma etch process, the method comprising:

-   -   predicting a first characteristic of a particle of a plasma in a        first scale based on a first plurality of parameters;    -   predicting a second characteristic of the particle in a second        scale based on a modification of the first characteristic caused        by a second plurality of parameters; and    -   simulating an etch characteristic of a feature based on the        first and the second characteristics of the particle.

43. The non-transitory computer readable medium of clause 42, whereinthe set of instructions that is executable by the one or more processorsof the apparatus to cause the apparatus to further perform:

-   -   predicting a sheath profile of the plasma in the first scale        based on the first plurality of parameters;    -   determining a gradient of the predicted sheath profile; and    -   determining an angle of incidence, a trajectory, or an energy of        the particle directed towards a wafer based on the gradient of        the predicted sheath profile.

44. The non-transitory computer readable medium of clause 43, whereinthe set of instructions that is executable by the one or more processorsof the apparatus to cause the apparatus to further perform:

-   -   accessing a layout of a die, the layout comprising a        pattern-perimeter density map; and    -   predicting the second characteristic of the particle of the        plasma based on the pattern-perimeter density map, wherein the        particle comprises a charged particle or an uncharged particle,        and wherein the second characteristic comprises a modification        of the angle of incidence, the trajectory, or the energy of the        particle.

45. The non-transitory computer readable medium of clause 44, whereinthe set of instructions that is executable by the one or more processorsof the apparatus to cause the apparatus to further perform:

-   -   identifying, based on the pattern-perimeter density map, a first        region of the die having a first pattern-perimeter density and a        second region of the die having a second pattern-perimeter        density different from the first pattern-perimeter density;    -   predicting an electric potential gradient based on the        identified first and the second regions; and    -   predicting the second characteristic of the charged particle        based on the electric potential gradient.

46. The non-transitory computer readable medium of clause 45, whereinthe set of instructions that is executable by the one or more processorsof the apparatus to cause the apparatus to further perform:

-   -   predicting a concentration gradient of an etchant between the        identified first and the second regions;    -   predicting a diffusion flux of the etchant based on the        concentration gradient; and    -   predicting the second characteristic of the uncharged particle        based on the diffusion flux.

47. The non-transitory computer readable medium of any one of clauses42-46, wherein the set of instructions that is executable by the one ormore processors of the apparatus to cause the apparatus to furtherperform predicting the second characteristic of the particle in thesecond scale based on a pattern-perimeter density gradient of the dieand a Gaussian kernel, wherein the Gaussian kernel is a multi-lengthscale kernel comprising a length scale ranging from 5 nm to 50 μm.

48. A non-transitory computer readable medium storing a set ofinstructions that is executable by one or more processors of anapparatus to cause the apparatus to perform a method of simulating aplasma etch process, the method comprising:

-   -   acquiring a first image of the feature;    -   identifying the feature based on pattern information;    -   predicting an etch profile of the feature to be etched using a        plasma etch process, the predicting comprising:        -   predicting a first characteristic of a particle of a plasma            in a first scale based on a first plurality of parameters;            and        -   predicting a second characteristic of the particle in a            second scale based on a modification of the first            characteristic caused by a second plurality of parameters;            and    -   generating a second image comprising the predicted etch profile        of the feature.

49. The non-transitory computer readable medium of clause 48, whereinthe set of instructions that is executable by the one or more processorsof the apparatus to cause the apparatus to further perform acquiring thefirst image from a user-defined database, wherein the user-defineddatabase comprises a graphic database system.

50. The non-transitory computer readable medium of any one of clauses 48and 49, wherein the pattern information comprises pattern-perimeterinformation, wherein the set of instructions that is executable by theone or more processors of the apparatus to cause the apparatus tofurther perform comparing the pattern-perimeter information and atrained feature from a trained image, using a machine learning network.

51. The non-transitory computer readable medium of any one of clauses48-50, wherein the set of instructions that is executable by the one ormore processors of the apparatus to cause the apparatus to furtherperform:

-   -   predicting a sheath profile of the plasma in the first scale        based on the first plurality of parameters;    -   determining a gradient of the predicted sheath profile; and    -   determining an angle of incidence, a trajectory, or an energy of        the particle directed towards a wafer based on the gradient of        the predicted sheath profile.

52. The non-transitory computer readable medium of clause 51, whereinthe set of instructions that is executable by the one or more processorsof the apparatus to cause the apparatus to further perform:

-   -   accessing a layout of a die, the layout comprising a        pattern-perimeter density map; and    -   predicting the second characteristic of the particle of the        plasma based on the pattern-perimeter density map, wherein the        particle comprises a charged particle or an uncharged particle,        and wherein the second characteristic comprises a modification        of the angle of incidence, the trajectory, or the energy of the        particle.

53. The non-transitory computer readable medium of clause 52, whereinthe set of instructions that is executable by the one or more processorsof the apparatus to cause the apparatus to further perform:

-   -   identifying, based on the pattern-perimeter density map, a first        region of the die having a first pattern-perimeter density and a        second region of the die having a second pattern-perimeter        density different from the first pattern-perimeter density;    -   predicting an electric potential gradient based on the        identified first and the second regions; and    -   predicting a modified trajectory of the charged particle based        on the electric potential gradient.

54. The non-transitory computer readable medium of any one of clauses48-53, wherein the set of instructions that is executable by the one ormore processors of the apparatus to cause the apparatus to furtherperform predicting the second characteristic of the particle in thesecond scale based on a pattern-perimeter density gradient of the dieand a Gaussian kernel, wherein the Gaussian kernel is a multi-lengthscale kernel comprising a length scale ranging from 5 nm to 50 μm.

55. A method for simulating a plasma etch process, the methodcomprising:

-   -   predicting, in a first scale, a first characteristic of a        chamber of a plurality of chambers configured to perform the        plasma etch process;    -   predicting, in a second scale, a second characteristic of the        chamber of the plurality of chambers, wherein the first scale        comprises the second scale; and    -   simulating an etch characteristic of a feature based on the        first and the second characteristics of the chamber.

56. The method of clause 55, further comprising:

-   -   predicting a first characteristic of a particle of a plasma in a        third scale based on a first plurality of parameters;    -   predicting a second characteristic of the particle in a fourth        scale based on a modification of the first characteristic of the        particle caused by a second plurality of parameters; and    -   simulating the etch characteristic of the feature based on the        first and the second characteristics of the particle.

57. The method of any one of clauses 55 and 56, wherein the firstcharacteristic of the chamber comprises a chamber status, a chambertype, or a chamber processing history, and wherein the secondcharacteristic of the chamber comprises a chamber wall condition, achamber pressure, or a characteristic of a focus ring of the chamber.

58. The method of clause 57, further comprising simulating the etchcharacteristic of the feature based on the characteristic of the focusring, wherein adjusting the characteristic of the focus ring adjusts thesimulated etch characteristic of the feature.

59. The method of clause 58, wherein adjusting the characteristic of thefocus ring comprises adjusting a position or an applied voltage to thefocus ring.

60. The method of any one of clauses 55-59, wherein the first scalecomprises a fab-scale and the second scale comprises a chamber-scale.

61. A plasma etch simulation system, comprising:

-   -   a memory storing a set of instructions; and    -   a processor configured to execute the set of instructions to        cause the plasma etch simulation system to:        -   predict, in a first scale, a first characteristic of a            chamber of a plurality of chambers configured to perform the            plasma etch process;        -   predict, in a second scale, a second characteristic of the            chamber of the plurality of chambers, wherein the first            scale comprises the second scale; and        -   simulate an etch characteristic of a feature based on the            first and the second characteristics of the chamber.

62. The system of clause 61, wherein the processor is configured toexecute the set of instructions to further cause the plasma etchsimulation system to:

-   -   predict a first characteristic of a particle of a plasma in a        third scale based on a first plurality of parameters;    -   predict a second characteristic of the particle in a fourth        scale based on a modification of the first characteristic of the        particle caused by a second plurality of parameters; and    -   simulate the etch characteristic of the feature based on the        first and the second characteristics of the particle.

63. The system of any one of clauses 61 and 62, wherein the firstcharacteristic of the chamber comprises a chamber status, a chambertype, or a chamber processing history, and wherein the secondcharacteristic of the chamber comprises a chamber wall condition, achamber pressure, or a characteristic of a focus ring of the chamber.

64. The system of clause 63, wherein an adjustment of the characteristicof the focus ring adjusts the simulated etch characteristic of thefeature.

65. The system of clause 64, wherein the adjustment of thecharacteristic of the focus ring comprises an adjustment of the positionor the applied voltage to the focus ring.

66. The system of any one of clauses 63 and 64, wherein thecharacteristic of the focus ring comprises a material, a position, anapplied voltage, or an operating condition of the focus ring.

The block diagrams in the figures illustrate the architecture,functionality, and operation of possible implementations of systems,methods, and computer hardware or software products according to variousexemplary embodiments of the present disclosure. In this regard, eachblock in a flowchart or block diagram may represent a module, segment,or portion of code, which comprises one or more executable instructionsfor implementing the specified logical functions. It should beunderstood that in some alternative implementations, functions indicatedin a block may occur out of the order noted in the figures. For example,two blocks shown in succession may be executed or implementedsubstantially concurrently, or two blocks may sometimes be executed inreverse order, depending upon the functionality involved. Some blocksmay also be omitted. It should also be understood that each block of theblock diagrams, and combination of the blocks, may be implemented byspecial purpose hardware-based systems that perform the specifiedfunctions or acts, or by combinations of special purpose hardware andcomputer instructions.

It will be appreciated that the embodiments of the present disclosureare not limited to the exact construction that has been described aboveand illustrated in the accompanying drawings, and that variousmodifications and changes may be made without departing from the scopethereof. The present disclosure has been described in connection withvarious embodiments, other embodiments of the invention will be apparentto those skilled in the art from consideration of the specification andpractice of the invention disclosed herein. It is intended that thespecification and examples be considered as exemplary only, with a truescope and spirit of the invention being indicated by the followingclaims.

The descriptions above are intended to be illustrative, not limiting.Thus, it will be apparent to one skilled in the art that modificationsmay be made as described without departing from the scope of the claimsset out below.

1. A non-transitory computer readable medium having instructions thereinor thereon, the instructions, when executed by one or more processors,configured to cause the one or more processors to at least: acquire afirst image of a feature; identify the feature based on patterninformation from the first image; predict an etch profile of the featureto be etched using a plasma etch process, the prediction comprising:prediction of a first characteristic of a particle of a plasma in afirst scale based on a first plurality of parameters; and prediction ofa second characteristic of the particle in a second scale based on amodification of the first characteristic caused by a second plurality ofparameters; and generation of a second image comprising the predictedetch profile of the feature.
 2. The medium of claim 1, wherein theinstructions are further configured to cause the one or more processorsto acquire the first image from a user-defined database, wherein theuser-defined database comprises a graphic database system.
 3. The mediumof claim 1, wherein the pattern information comprises pattern-perimeterinformation, and wherein the instructions configured to cause the one ormore processors to identify the feature are further configured to causethe one or more processors to compare the pattern-perimeter informationand a trained feature from a trained image, using a machine learningnetwork.
 4. The medium of claim 1, wherein the first image comprises anafter-develop image of the feature.
 5. The medium of claim 1, whereinthe first scale comprises a wafer-scale and the second scale comprises adie-scale.
 6. The medium of claim 5, wherein the instructions configuredto cause the one or more processors to predict the etch profile arefurther configured to cause the one or more processors to predict asheath profile of the plasma in the wafer-scale based on the firstplurality of parameters.
 7. The medium of claim 6, wherein theinstructions configured to cause the one or more processors to predictthe first characteristic are further configured to cause the one or moreprocessors to determine a gradient of the predicted sheath profile, andwherein the first characteristic comprises an angle of incidence, atrajectory, or an energy of the particle directed towards a wafer. 8.The medium of claim 7, wherein the first plurality of parameterscomprises geometry of a plasma reactor configured to perform the plasmaetch process, a process condition for the plasma etch process, or alocation on the wafer.
 9. The medium of claim 7, wherein theinstructions configured to cause the one or more processors to predictthe second characteristic are further configured to cause the one ormore processors to predict a modification of the angle of incidence, thetrajectory, or the energy of the particle in the die-scale.
 10. Themedium of claim 9, wherein the instructions configured to cause the oneor more processors to predict the second characteristic are furtherconfigured to cause the one or more processors to: access a layout of adie, the layout comprising a pattern-perimeter density map; and predictthe second characteristic of the particle based on the pattern-perimeterdensity map.
 11. The medium of claim 10, wherein the particle comprisesa charged particle and the instructions configured to cause the one ormore processors to predict the second characteristic of the chargedparticle of the plasma are further configured to cause the one or moreprocessors to: identify, based on the pattern-perimeter density map, afirst region of the die having a first pattern-perimeter density and asecond region of the die having a second pattern-perimeter densitydifferent from the first pattern-perimeter density; predict an electricpotential gradient based on the identified first and second regions; andpredict the second characteristic of the charged particle based on theelectric potential gradient.
 12. The medium of claim 11, wherein theinstructions configured to cause the one or more processors to predictthe electric potential gradient are further configured to cause the oneor more processors to: predict a first electric potential of the firstregion comprising a dense region having a high pattern-perimeterdensity; and predict a second electric potential of the second regioncomprising an isolated region having a lower pattern-perimeter density,wherein the first and the second electric potentials are different. 13.The medium of claim 10, wherein the particle comprises an unchargedparticle and the instructions configured to cause the one or moreprocessors to predict the second characteristic of the unchargedparticle of the plasma are further configured to cause the one or moreprocessors to: identify, based on the pattern-perimeter density map, afirst region of the die having a first pattern-perimeter density and asecond region of the die having a second pattern-perimeter densitydifferent from the first pattern-perimeter density; predict aconcentration gradient of an etchant between the identified first andthe second regions; predict a diffusion flux of the etchant based on theconcentration gradient; and predict the second characteristic of theuncharged particle based on the diffusion flux.
 14. The medium of claim13, wherein the instructions configured to cause the one or moreprocessors to predict the concentration gradient are further configuredto cause the one or more processors to: predict a first concentration ofthe etchant in the first region; and predict a second concentration ofthe etchant in the second region, wherein the first and secondconcentrations are different.
 15. The medium of claim 9, wherein theinstructions are further configured to cause the one or more processorsto predict the second characteristic of the particle in the second scalebased on a pattern-perimeter density gradient of the die and a Gaussiankernel, wherein the Gaussian kernel is a multi-length scale kernelcomprising a length scale ranging from 5 nm to 50 pm, and wherein thesecond plurality of parameters comprises the layout, a pattern-perimeterdensity; a pattern-perimeter density variation, or the pattern-perimeterdensity gradient of the die.
 16. A method comprising: acquiring a firstimage of the feature; identifying the feature based on apattern-perimeter information from the image; predicting, by a hardwarecomputer system, an etch profile of the feature to be etched using aplasma etch process, the predicting comprising: predicting a firstcharacteristic of a particle of a plasma in a first scale based on afirst plurality of parameters; and predicting a second characteristic ofthe particle in a second scale based on a modification of the firstcharacteristic caused by a second plurality of parameters; andgenerating a second image comprising the predicted etch profile of thefeature.
 17. The method of claim 16, further comprising acquiring thefirst image from a user-defined database, wherein the user-defineddatabase comprises a graphic database system.
 18. The method of claim16, wherein the identifying the feature comprises comparing thepattern-perimeter information and a trained feature from a trainedimage, using a machine learning network.
 19. The method of claim 16,wherein the first image comprises an after-develop image of the feature.20. The method of claim 16, wherein the first scale comprises awafer-scale and the second scale comprises a die-scale.